Triangle calculation
The following list contains most of the known formulas from trigonometry in the plane . Most of these relationships use trigonometric functions .
The following terms are used: The triangle has the sides , and , the angles , and at the corners , and . Furthermore, let the radius of the radius , the Inkreisradius and , and the Ankreisradien (namely, the radii of the excircles, the corners , or opposite) of the triangle . The variable stands for half the circumference of the triangle :
![ABC](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e55b44cfd965fbdc7a328d5db8a35a619db0971)
![a = BC](https://wikimedia.org/api/rest_v1/media/math/render/svg/944ad306da7031c91d6293d105e1fcc6fdb87b47)
![b = CA](https://wikimedia.org/api/rest_v1/media/math/render/svg/5914a6e34b4d6a947cf7bd1e1234018329899ceb)
![\alpha](https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3)
![\beta](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![B.](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
![C.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029)
![r](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538)
![\ rho](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64)
![\ rho _ {a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf5bcd5f18e417072a2f15baf57774af83e76672)
![\ rho _ {b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e71b4c1f19dae23cb553215f66d17cdbe4853dca)
![\ rho _ {c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/523e8c709cfeaef7ed497ce5be80144fdf2e9b70)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![B.](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
![C.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029)
![ABC](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e55b44cfd965fbdc7a328d5db8a35a619db0971)
![s](https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632)
![ABC](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e55b44cfd965fbdc7a328d5db8a35a619db0971)
-
.
Finally, the area of the triangle is with designated. All other terms are explained in the relevant sections in which they appear.
![ABC](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e55b44cfd965fbdc7a328d5db8a35a619db0971)
![F.](https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57)
It should be noted here that the names for the perimeter radius , the Inkreisradius and the three Ankreisradien , , be used. Often notwithstanding, for the same sizes and the names , , , , used.
![r](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538)
![\ rho](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64)
![\ rho _ {a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf5bcd5f18e417072a2f15baf57774af83e76672)
![\ rho _ {b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e71b4c1f19dae23cb553215f66d17cdbe4853dca)
![\ rho _ {c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/523e8c709cfeaef7ed497ce5be80144fdf2e9b70)
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![r](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538)
![r_a](https://wikimedia.org/api/rest_v1/media/math/render/svg/da5f0781a46ba20129a554237056b6ade78b956f)
![r_ {b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c22eac9d11ab84da51b891d29ee2e5de75eab1a)
![r_ {c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e41dc9dcda9fd0c224d94555361f02b73aacd2e)
Angle sum
![\ alpha + \ beta + \ gamma = 180 ^ {\ circ}](https://wikimedia.org/api/rest_v1/media/math/render/svg/720f59b296cd42f5d2968b211dd56394a860c97a)
Formula 1:
![{\ frac {a} {\ sin \ alpha}} = {\ frac {b} {\ sin \ beta}} = {\ frac {c} {\ sin \ gamma}} = 2r = {\ frac {abc} {2F}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b9f3e7bc934d0ede764bf31b2e1a0d4ec288e64)
Formula 2:
if
![\ sin \ beta = {\ frac {b} {a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/59d3b6233780d0dfb64cb62df635f935da9b7ebc)
![\ sin \ gamma = \ frac {c} {a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a0930aabd4b02f54f5a9d4bb1b7d72b077797a2)
if
![\ sin \ alpha = \ frac {a} {b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1747314f67cce94eadaa090d0dc863b12522c21)
![\ sin \ gamma = \ frac {c} {b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1eb012ae1c2c877db8b2dfae670d32d924cf84b)
if
![\ sin \ alpha = \ frac {a} {c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30a3bd270353e0f19f4f16054ac368730cbdc4d6)
![\ sin \ beta = \ frac {b} {c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/efbb10656691c5f28b422df92501cad81c2b45f1)
Formula 1:
![a ^ {2} = b ^ {2} + c ^ {2} -2bc \ \ cos \ alpha](https://wikimedia.org/api/rest_v1/media/math/render/svg/e45e1928a8a7a05f206c821b17cdd92a37a253fb)
![b ^ {2} = c ^ {2} + a ^ {2} -2ca \ \ cos \ beta](https://wikimedia.org/api/rest_v1/media/math/render/svg/1fb073ee8b8aa66a405deba589315d95f457d700)
![c ^ {2} = a ^ {2} + b ^ {2} -2ab \ \ cos \ gamma](https://wikimedia.org/api/rest_v1/media/math/render/svg/23935d3526342e0d4919166fbe9d703fb1251d0f)
Formula 2:
if
![{\ displaystyle \ cos \ beta = {\ frac {c} {a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/750e9a7e7b661c67cd5ea7bdbd96487bfa70748b)
![{\ displaystyle \ cos \ gamma = {\ frac {b} {a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c96eccc19f8aa6d899f0efdfd0fb0b8ad56d67f)
if
![{\ displaystyle \ cos \ alpha = {\ frac {c} {b}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01db57dd79cfdb8e2c6a389d171225ba0dc1d196)
![{\ displaystyle \ cos \ gamma = {\ frac {a} {b}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1ae8c94d0b91a6bdf7ef0542828bcd91b5636df)
if
-
( Pythagorean theorem )
![{\ displaystyle \ cos \ alpha = {\ frac {b} {c}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42ab6e3e280c970df88e4c7764510cda3fa4082e)
![{\ displaystyle \ cos \ beta = {\ frac {a} {c}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5daa42754216029f8181ebee80e49c6bd8c30c4d)
Projection set
![a = b \, \ cos \ gamma + c \, \ cos \ beta](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f276ede56e0ac902b19006ebb5dc78b9e09a4a3)
![b = c \, \ cos \ alpha + a \, \ cos \ gamma](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b6cb8d834b6b741de3c681bf5ce922962a52067)
![c = a \, \ cos \ beta + b \, \ cos \ alpha](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a4b31b716e5379d0b5293f1144a3b78c798317d)
![{\ displaystyle {\ frac {b + c} {a}} = {\ frac {\ cos {\ frac {\ beta - \ gamma} {2}}} {\ sin {\ frac {\ alpha} {2} }}}, \ quad {\ frac {c + a} {b}} = {\ frac {\ cos {\ frac {\ gamma - \ alpha} {2}}} {\ sin {\ frac {\ beta} {2}}}}, \ quad {\ frac {a + b} {c}} = {\ frac {\ cos {\ frac {\ alpha - \ beta} {2}}} {\ sin {\ frac { \ gamma} {2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c12cfe54ce4f4742bfc388c37b8b5c4a3072633d)
![{\ displaystyle {\ frac {bc} {a}} = {\ frac {\ sin {\ frac {\ beta - \ gamma} {2}}} {\ cos {\ frac {\ alpha} {2}}} }, \ quad {\ frac {ca} {b}} = {\ frac {\ sin {\ frac {\ gamma - \ alpha} {2}}} {\ cos {\ frac {\ beta} {2}} }}, \ quad {\ frac {ab} {c}} = {\ frac {\ sin {\ frac {\ alpha - \ beta} {2}}} {\ cos {\ frac {\ gamma} {2} }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce914ac7c3ee93563308c729d1708c56c2feca4f)
Formula 1:
![\ frac {b + c} {bc} = \ frac {\ tan \ frac {\ beta + \ gamma} {2}} {\ tan \ frac {\ beta - \ gamma} {2}} = \ frac {\ cot \ frac {\ alpha} {2}} {\ tan \ frac {\ beta - \ gamma} {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdd7ff080521d78d9889ea0822966194a1dda2d9)
Analogous formulas apply to and :
![{\ frac {a + b} {ab}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93415740a91d0b1807091264dae0b6d854ff4e65)
![{\ displaystyle {\ frac {c + a} {ca}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08af94c0c2fd7256196b86783eb70abc191b3284)
![{\ frac {a + b} {ab}} = {\ frac {\ tan {\ frac {\ alpha + \ beta} {2}}} {\ tan {\ frac {\ alpha - \ beta} {2} }}} = {\ frac {\ cot {\ frac {\ gamma} {2}}} {\ tan {\ frac {\ alpha - \ beta} {2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ab81d0ef0db556f0300e99a4404d152994ff164)
![{\ displaystyle {\ frac {c + a} {ca}} = {\ frac {\ tan {\ frac {\ gamma + \ alpha} {2}}} {\ tan {\ frac {\ gamma - \ alpha} {2}}}} = {\ frac {\ cot {\ frac {\ beta} {2}}} {\ tan {\ frac {\ gamma - \ alpha} {2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/61704ca0719701915d27e8c9762b91998e9ae889)
Because of this , one of these formulas remains valid if both the sides and the associated angles are swapped, for example:
![{\ displaystyle \ tan (-x) = - \ tan (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/961705989441ed1a5aa05bf810320d4851b2db32)
![{\ frac {a + c} {ac}} = {\ frac {\ tan {\ frac {\ alpha + \ gamma} {2}}} {\ tan {\ frac {\ alpha - \ gamma} {2} }}} = {\ frac {\ cot {\ frac {\ beta} {2}}} {\ tan {\ frac {\ alpha - \ gamma} {2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5038d864c738e25fa32ff93a9c66bfce67c4dd21)
Formula 2:
if
![\ tan \ beta = \ frac {b} {c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9260b12985d1251fb5ff22c8d45e50d5130760e2)
![\ tan \ gamma = \ frac {c} {b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c176a3c55e83a1c0d8e11d298700a7362a6a2c98)
if
![\ tan \ alpha = \ frac {a} {c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f65fabc9497ebb14b6751877a91ea537c67f26b5)
![\ tan \ gamma = \ frac {c} {a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08ad06bf52c65829acac3804c59ba83762326ae3)
if
![\ tan \ alpha = \ frac {a} {b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6190698fd29dcde8ede380d821a63b2d4bb16f45)
![\ tan \ beta = \ frac {b} {a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1320d9c8aab9ff714e0215f9a930334662218b2)
Formulas with half the circumference
Below is always half the circumference of the triangle , that is .
![s](https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632)
![ABC](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e55b44cfd965fbdc7a328d5db8a35a619db0971)
![s = {\ frac {a + b + c} {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/787a98dac5681f383514fc1bd5b4d8e561a3fd21)
![sa = \ frac {b + ca} {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/805906b78313058f280e780574d0f3bcbfb8b020)
![sb = \ frac {c + ab} {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6bd7bde06fbc044cb9730d7d947725402708a94)
![sc = \ frac {a + bc} {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/140397740da1e53d188a32ebc8a4f399f2e74cfb)
![\ left (sb \ right) + \ left (sc \ right) = a](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b790e361a7dd45b6ae342b0c7ed87b7bf6238ef)
![\ left (sc \ right) + \ left (sa \ right) = b](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4c9cc35cf083c5cecf5f4608d128e21a33ea4cb)
![\ left (sa \ right) + \ left (sb \ right) = c](https://wikimedia.org/api/rest_v1/media/math/render/svg/81f17eccd84980303069f12ee45ab3c62310fc0a)
![\ left (sa \ right) + \ left (sb \ right) + \ left (sc \ right) = s](https://wikimedia.org/api/rest_v1/media/math/render/svg/4377759001491bc6f483eb692990de8f36e311a9)
![\ sin \ frac {\ alpha} {2} = \ sqrt {\ frac {\ left (sb \ right) \ left (sc \ right)} {bc}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a447976eb599b2d1fe0a5147926c9cd2c238687)
![{\ displaystyle \ sin {\ frac {\ beta} {2}} = {\ sqrt {\ frac {\ left (sc \ right) \ left (sa \ right)} {ca}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2e189984b34d3d699ecc84f7cc9e2227305dba2)
![\ sin \ frac {\ gamma} {2} = \ sqrt {\ frac {\ left (sa \ right) \ left (sb \ right)} {ab}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac3176b303461e46a5f812dbaab293791127d501)
![\ cos \ frac {\ alpha} {2} = \ sqrt {\ frac {s \ left (sa \ right)} {bc}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9077a7f58c3b1ef468cf4507f72f3bbc40d826f8)
![\ cos \ frac {\ beta} {2} = \ sqrt {\ frac {s \ left (sb \ right)} {ca}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c6d5ca6641f397070a46ce77d90ecdd09afe1f7)
![\ cos \ frac {\ gamma} {2} = \ sqrt {\ frac {s \ left (sc \ right)} {ab}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0879c030799bd86dd233826620d9c09f86b18c20)
![\ tan \ frac {\ alpha} {2} = \ sqrt {\ frac {\ left (sb \ right) \ left (sc \ right)} {s \ left (sa \ right)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ae1d36d5484cdd5cbd5f92ec2d4939e384c576b)
![{\ displaystyle \ tan {\ frac {\ beta} {2}} = {\ sqrt {\ frac {\ left (sc \ right) \ left (sa \ right)} {s \ left (sb \ right)}} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2a38d91c436219140883746ca8fe45a9814442d)
![\ tan \ frac {\ gamma} {2} = \ sqrt {\ frac {\ left (sa \ right) \ left (sb \ right)} {s \ left (sc \ right)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/113f756b9fd39c502412ab4baa0839d1fbef440a)
![s = 4r \ cos \ frac {\ alpha} {2} \ cos \ frac {\ beta} {2} \ cos \ frac {\ gamma} {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/587b54719bb9e19b0b0b5b426e7f78b85afa8e0d)
![sa = 4r \ cos \ frac {\ alpha} {2} \ sin \ frac {\ beta} {2} \ sin \ frac {\ gamma} {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1910018847cb4187c26654acd4ee77eb627c41f)
Area and radius
The area of the triangle is denoted here with (not, as is common today, with , in order to avoid confusion with the triangle corner):
![F.](https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
Heron's formula:
![F = \ sqrt {s \ left (sa \ right) \ left (sb \ right) \ left (sc \ right)} = \ frac {1} {4} \ sqrt {\ left (a + b + c \ right ) \ left (b + ca \ right) \ left (c + ab \ right) \ left (a + bc \ right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1cff40f9e76f4a6caedc61ec2ad061494109357d)
![F = \ frac {1} {4} \ sqrt {2 \ left (b ^ {2} c ^ {2} + c ^ {2} a ^ {2} + a ^ {2} b ^ {2} \ right) - \ left (a ^ {4} + b ^ {4} + c ^ {4} \ right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dfd7d8e89014c6445ec227181353f4812439c1a7)
Further area formulas:
![F = \ frac {1} {2} bc \ sin \ alpha = \ frac {1} {2} ca \ sin \ beta = \ frac {1} {2} from \ sin \ gamma](https://wikimedia.org/api/rest_v1/media/math/render/svg/b112157af09c0eceebb7617060f228b7c04532ad)
-
Wherein , and the lengths of from , or outgoing heights of the triangle are.![Ha}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00531874c3a9e751e1c78d4f84483fcec2e75eba)
![h_ {b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/140aec52ffb1fd0966772704b2fe00827cdefa13)
![h_ {c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d06972075e10d9390c826454530d3e2a6351dc45)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![B.](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
![C.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029)
![ABC](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e55b44cfd965fbdc7a328d5db8a35a619db0971)
![F = 2 r ^ {2} \ sin \, \ alpha \, \ sin \, \ beta \, \ sin \, \ gamma](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf54fc56018410ebe963dba45ff0a9d863466bb8)
![F = \ frac {abc} {4r}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e02db2024cb4f6fc1f1f4ee4796191defd934ff)
![F = \ rho s = \ rho_ {a} \ left (sa \ right) = \ rho _ {b} \ left (sb \ right) = \ rho_ {c} \ left (sc \ right)](https://wikimedia.org/api/rest_v1/media/math/render/svg/23334167893cfcb4378336e8a78ce99f6dc6faf8)
![F = \ sqrt {\ rho \ rho _ {a} \ rho _ {b} \ rho _ {c}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/448f20368abca205d9232128a79409234d18674e)
![F = 4 \ rho r \ cos \, \ frac {\ alpha} {2} \, \ cos \, \ frac {\ beta} {2} \, \ cos \, \ frac {\ gamma} {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b905a9c2047cf11c02f4dd52f5536804918ac5e7)
![F = s ^ {2} \ tan \, \ frac {\ alpha} {2} \, \ tan \, \ frac {\ beta} {2} \, \ tan \, \ frac {\ gamma} {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2cb2fcf6dddc8afc9680b294cd3c7f48a7364d3a)
-
, With
![{\ displaystyle F = {\ sqrt {\ dfrac {r \, h_ {a} \, h_ {b} \, h_ {c}} {2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65d2c59e52dc9892f8e396d749d3c3f93b81263d)
![{\ displaystyle F = {\ dfrac {\, h_ {a} \, h_ {b} \, h_ {c}} {2 \ rho \, {(\ sin \ alpha + \ sin \ beta + \ sin \ gamma )}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f33b324dbec9587339c400f1ff9b9dad6f6c30ea)
Extended sine law:
![a = 2 r \, \ sin \ alpha](https://wikimedia.org/api/rest_v1/media/math/render/svg/109c21c9dfe7128ff5b7f4e15a19eeb5f3f85f16)
![b = 2 r \, \ sin \ beta](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f643cdbf10e006ad55c3750bcd7206012a37b0b)
![c = 2 r \, \ sin \ gamma](https://wikimedia.org/api/rest_v1/media/math/render/svg/66c8ae7a50c576a6f38c8161f79582a5ec43ed40)
![r = \ frac {abc} {4F}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8bebd59edb3ce6524e3f7763493ceebf0095eb90)
Inside and circle radii
This section lists formulas in which the incircle radius , the circle radius , and the triangle appear.
![\ rho _ {a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf5bcd5f18e417072a2f15baf57774af83e76672)
![\ rho _ {b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e71b4c1f19dae23cb553215f66d17cdbe4853dca)
![\ rho _ {c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/523e8c709cfeaef7ed497ce5be80144fdf2e9b70)
![ABC](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e55b44cfd965fbdc7a328d5db8a35a619db0971)
![\ rho = \ left (sa \ right) \ tan \ frac {\ alpha} {2} = \ left (sb \ right) \ tan \ frac {\ beta} {2} = \ left (sc \ right) \ tan \ frac {\ gamma} {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/123af333044eb976dc0e48f1581eaf56d0f8d6ce)
![\ rho = 4r \ sin \ frac {\ alpha} {2} \ sin \ frac {\ beta} {2} \ sin \ frac {\ gamma} {2} = s \ tan \ frac {\ alpha} {2} \ tan \ frac {\ beta} {2} \ tan \ frac {\ gamma} {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9857afda4eafd324f3213cb408fc05d99c47a43f)
![\ rho = r \ left (\ cos \ alpha + \ cos \ beta + \ cos \ gamma -1 \ right)](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6f715fb9c42db9054cd8e4bb97246f539e8c769)
![\ rho = \ frac {F} {s} = \ frac {abc} {4rs}](https://wikimedia.org/api/rest_v1/media/math/render/svg/675eb7b0cc7865d9cc78406795fb8dfd6d03d40a)
![\ rho = \ sqrt {\ frac {\ left (sa \ right) \ left (sb \ right) \ left (sc \ right)} {s}} = \ frac {1} {2} \ sqrt {\ frac { \ left (b + ca \ right) \ left (c + ab \ right) \ left (a + bc \ right)} {a + b + c}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ee6be8d1a210441f6780900cb46638a2e1c5324)
![\ rho = \ frac {a} {\ cot \ frac {\ beta} {2} + \ cot \ frac {\ gamma} {2}} = \ frac {b} {\ cot \ frac {\ gamma} {2 } + \ cot \ frac {\ alpha} {2}} = \ frac {c} {\ cot \ frac {\ alpha} {2} + \ cot \ frac {\ beta} {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f60025d349a8308f78dac9fdf72833529b9b30d)
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Major inequality ; Equality only occurs when triangle is equilateral.
![2 \ rho \ leq r](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7ea525418062bd1fa337a3e937d2629bfda73c8)
![ABC](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e55b44cfd965fbdc7a328d5db8a35a619db0971)
![\ rho _ {a} = s \ tan \ frac {\ alpha} {2} = \ left (sb \ right) \ cot \ frac {\ gamma} {2} = \ left (sc \ right) \ cot \ frac {\ beta} {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74243f4ab0fef960ccff4ffe96d65b39e69e9230)
![\ rho _ {a} = 4r \ sin \ frac {\ alpha} {2} \ cos \ frac {\ beta} {2} \ cos \ frac {\ gamma} {2} = \ left (sa \ right) \ tan \ frac {\ alpha} {2} \ cot \ frac {\ beta} {2} \ cot \ frac {\ gamma} {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fae2e5f2043111b9fd0fe71e27fa1a286519e2fc)
![\ rho _ {a} = r \ left (- \ cos \ alpha + \ cos \ beta + \ cos \ gamma +1 \ right)](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec8ff598946bbe8b846a2207df8e4b7e97f29cf5)
![\ rho _ {a} = \ frac {F} {sa} = \ frac {abc} {4r \ left (sa \ right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ecb9a6c0fc9888c379855bc1b989da34623ba433)
![\ rho _ {a} = \ sqrt {\ frac {s \ left (sb \ right) \ left (sc \ right)} {sa}} = \ frac {1} {2} \ sqrt {\ frac {\ left (a + b + c \ right) \ left (c + ab \ right) \ left (a + bc \ right)} {b + ca}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5de20d0996116d3ba61b7aa8c750cc35581b168e)
The circles are equal: Each formula for applies analogously to and .
![\ rho _ {a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf5bcd5f18e417072a2f15baf57774af83e76672)
![\ rho _ {b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e71b4c1f19dae23cb553215f66d17cdbe4853dca)
![\ rho _ {c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/523e8c709cfeaef7ed497ce5be80144fdf2e9b70)
![\ frac {1} {\ rho} = \ frac {1} {\ rho _ {a}} + \ frac {1} {\ rho _ {b}} + \ frac {1} {\ rho _ {c} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/65849660605ed119305465c8d5b13d1b569d4767)
Heights
The lengths of from , or outgoing heights of the triangle are with , and referred to.
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![B.](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
![C.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029)
![ABC](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e55b44cfd965fbdc7a328d5db8a35a619db0971)
![Ha}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00531874c3a9e751e1c78d4f84483fcec2e75eba)
![h_ {b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/140aec52ffb1fd0966772704b2fe00827cdefa13)
![h_ {c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d06972075e10d9390c826454530d3e2a6351dc45)
![h _ {{a}} = b \ sin \ gamma = c \ sin \ beta = {\ frac {2F} {a}} = 2r \ sin \ beta \ sin \ gamma = 2r \ left (\ cos \ alpha + \ cos \ beta \ cos \ gamma \ right)](https://wikimedia.org/api/rest_v1/media/math/render/svg/d688efbcd9860b7fbd8d4e3fc7ffad735de9f904)
![h _ {{b}} = c \ sin \ alpha = a \ sin \ gamma = {\ frac {2F} {b}} = 2r \ sin \ gamma \ sin \ alpha = 2r \ left (\ cos \ beta + \ cos \ alpha \ cos \ gamma \ right)](https://wikimedia.org/api/rest_v1/media/math/render/svg/f37d2bbbcc94914e8fe55c8ae3e7e6c135cccfce)
![h _ {{c}} = a \ sin \ beta = b \ sin \ alpha = {\ frac {2F} {c}} = 2r \ sin \ alpha \ sin \ beta = 2r \ left (\ cos \ gamma + \ cos \ alpha \ cos \ beta \ right)](https://wikimedia.org/api/rest_v1/media/math/render/svg/4da5c421d1cc29f56d5fc3ad546fe0c205de1a1b)
![h_ {a} = \ frac {a} {\ cot \ beta + \ cot \ gamma}; \; \; \; \; \; h_ {b} = \ frac {b} {\ cot \ gamma + \ cot \ alpha}; \; \; \; \; \; h_ {c} = \ frac {c} {\ cot \ alpha + \ cot \ beta}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ade245e6f1d669d9ca12e246bcf43f99fa3e6503)
![F = \ frac {1} {2} ah_ {a} = \ frac {1} {2} bh_ {b} = \ frac {1} {2} ch_ {c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4399ece46d68e533b08a4075b3e6e6fc5a34540)
![\ frac {1} {h_ {a}} + \ frac {1} {h_ {b}} + \ frac {1} {h_ {c}} = \ frac {1} {\ rho} = \ frac {1 } {\ rho _ {a}} + \ frac {1} {\ rho _ {b}} + \ frac {1} {\ rho _ {c}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b0d8fa4ea77d26e2e223a1f9fa927e0cf012883)
If the triangle has a right angle at (is therefore ), then applies
![ABC](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e55b44cfd965fbdc7a328d5db8a35a619db0971)
![C.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029)
![\ gamma = 90 ^ {\ circ}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08ed45556d3a8174ad5d5a4b3f9e0451024aba9c)
![h_ {c} = \ frac {ab} {c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb7d5860452f774d25ff0893d4ca96357fe222b0)
![{\ displaystyle h_ {a} = b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27c903b2c7ce86550f305e74a56cc9f9e75e1db3)
![{\ displaystyle h_ {b} = a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b15d371c92a6fee23e02e93e337c67393e04822)
Bisector
The lengths of from , or outgoing medians of the triangle is , and called.
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![B.](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
![C.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029)
![ABC](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e55b44cfd965fbdc7a328d5db8a35a619db0971)
![s_ {a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdf3b2fb584c60084dba330329b2ebc2982d9b88)
![s_ {b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de2f6218809aff65f4c3ed8bdf0f7bed2dc393a1)
![s_ {c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91c4d1b073949e8f032e50ff913afbf43bb5acb7)
![s_ {a} = \ frac {1} {2} \ sqrt {2b ^ {2} + 2c ^ {2} -a ^ {2}} = \ frac {1} {2} \ sqrt {b ^ {2 } + c ^ {2} + 2bc \ cos \ alpha} = \ sqrt {\ frac {a ^ {2}} {4} + bc \ cos \ alpha}](https://wikimedia.org/api/rest_v1/media/math/render/svg/289862303972afe17e27017257f12c8f29de36f0)
![s_ {b} = \ frac {1} {2} \ sqrt {2c ^ {2} + 2a ^ {2} -b ^ {2}} = \ frac {1} {2} \ sqrt {c ^ {2 } + a ^ {2} + 2ca \ cos \ beta} = \ sqrt {\ frac {b ^ {2}} {4} + ca \ cos \ beta}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f4a3fa3157bb899b6b967b614ba18b5787ad827)
![s_ {c} = \ frac {1} {2} \ sqrt {2a ^ {2} + 2b ^ {2} -c ^ {2}} = \ frac {1} {2} \ sqrt {a ^ {2 } + b ^ {2} + 2ab \ cos \ gamma} = \ sqrt {\ frac {c ^ {2}} {4} + ab \ cos \ gamma}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb28fa025ddd498456e0db9b3f5fdaa99b428c3b)
![s_ {a} ^ {2} + s_ {b} ^ {2} + s_ {c} ^ {2} = \ frac {3} {4} \ left (a ^ {2} + b ^ {2} + c ^ {2} \ right)](https://wikimedia.org/api/rest_v1/media/math/render/svg/b610513559381f16639b625b310cd391d6932126)
Bisector
We denote by , and the lengths of the of , or outgoing bisecting the triangle .
![w_ \ alpha](https://wikimedia.org/api/rest_v1/media/math/render/svg/5138271001e8db3f92ad1431b891a2d1434fe150)
![w _ {\ beta}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b23a4aaa3542828081105d33840517fec3a35db)
![w _ {\ gamma}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91ed8aac2720d782abc0fef7058a0c7b98a03f36)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![B.](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
![C.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029)
![ABC](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e55b44cfd965fbdc7a328d5db8a35a619db0971)
![{\ displaystyle w _ {\ alpha} = {\ frac {2bc \ cos {\ frac {\ alpha} {2}}} {b + c}} = {\ frac {2F} {a \ cos {\ frac {\ beta - \ gamma} {2}}}} = {\ frac {\ sqrt {bc (b + ca) (a + b + c)}} {b + c}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf920ff7c11a3e7861045cbc6c7889e74e1743b)
![{\ displaystyle w _ {\ beta} = {\ frac {2ca \ cos {\ frac {\ beta} {2}}} {c + a}} = {\ frac {2F} {b \ cos {\ frac {\ gamma - \ alpha} {2}}}} = {\ frac {\ sqrt {ca (c + ab) (a + b + c)}} {c + a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e69561c06ed8800d722956e0eeec762252e155ea)
![{\ displaystyle w _ {\ gamma} = {\ frac {2ab \ cos {\ frac {\ gamma} {2}}} {a + b}} = {\ frac {2F} {c \ cos {\ frac {\ alpha - \ beta} {2}}}} = {\ frac {\ sqrt {ab (a + bc) (a + b + c)}} {a + b}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1fadac3f992052c3db9b0f8f13d8630213ff9168)
General trigonometry in the plane
periodicity
![{\ displaystyle \ sin x \ quad = \ quad \ sin (x + 2n \ pi); \ quad n \ in \ mathbb {Z}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9a3490c5417b44bf994273df2a39a3d5acb9d16)
![{\ displaystyle \ cos x \ quad = \ quad \ cos (x + 2n \ pi); \ quad n \ in \ mathbb {Z}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5642c5006b7c71131c6e5a4f640d4c88900fb70c)
![{\ displaystyle \ tan x \ quad = \ quad \ tan (x + n \ pi); \ quad n \ in \ mathbb {Z}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c11437247b42bbc96b3fc7c3fde79824f719644)
![{\ displaystyle \ cot x \ quad = \ quad \ cot (x + n \ pi); \ quad n \ in \ mathbb {Z}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b181225746f33b8d53c59afad3c08ec6798f5dbf)
Mutual representation
The trigonometric functions can be converted into one another or represented mutually. The following relationships apply:
![\ tan x = \ frac {\ sin x} {\ cos x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f024fc11e57e7fbac757210cc79be66aa12ab86b)
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(" Trigonometric Pythagoras ")
![1+ \ tan ^ {2} x = \ frac {1} {\ cos ^ {2} x} = \ sec ^ {2} x](https://wikimedia.org/api/rest_v1/media/math/render/svg/e450d1ca2a9212ea5780105dbe047441cc29c713)
![1+ \ cot ^ {2} x = \ frac {1} {\ sin ^ {2} x} = \ csc ^ {2} x](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a198c3d1942fcf14c03cfd93f7e19a993bc6c1f)
(See also the section on phase shifts .)
Using these equations, the three functions that occur can be represented by one of the other two:
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Sign of the trigonometric functions
![\ sin x> 0 \ quad {\ text {for}} \ quad x \ in \ left] 0 ^ {{\ circ}}, 180 ^ {\ circ} \ right [](https://wikimedia.org/api/rest_v1/media/math/render/svg/fdac0f290fd61f2d4f5b9a9696bf3169b7cd0918)
![\ sin x <0 \ quad {\ text {for}} \ quad x \ in \ left] 180 ^ {{\ circ}}, 360 ^ {\ circ} \ right [](https://wikimedia.org/api/rest_v1/media/math/render/svg/170b6c1d4a0f5062cb5c97c5c1157b4e2535e625)
![\ cos x> 0 \ quad {\ text {for}} \ quad x \ in \ left [0 ^ {\ circ}, 90 ^ {\ circ} \ right [\ cup \ left] 270 ^ {\ circ}, 360 ^ {\ circ} \ right]](https://wikimedia.org/api/rest_v1/media/math/render/svg/af94605331a13a2721b46d99396163d49faf800d)
![\ cos x <0 \ quad {\ text {for}} \ quad x \ in \ left] 90 ^ {{\ circ}}, 270 ^ {\ circ} \ right [](https://wikimedia.org/api/rest_v1/media/math/render/svg/518e3b67bc5d734254ae99debc89eb0aee804f3e)
![\ tan x> 0 \ quad {\ text {for}} \ quad x \ in \ left] 0 ^ {{\ circ}}, 90 ^ {\ circ} \ right [\ cup \ left] 180 ^ {\ circ }, 270 ^ {\ circ} \ right [](https://wikimedia.org/api/rest_v1/media/math/render/svg/28fa7ed5c5b9ee66db9fa769dd73c51984ab1557)
![\ tan x <0 \ quad {\ text {for}} \ quad x \ in \ left] 90 ^ {{\ circ}}, 180 ^ {\ circ} \ right [\ cup \ left] 270 ^ {\ circ }, 360 ^ {\ circ} \ right [](https://wikimedia.org/api/rest_v1/media/math/render/svg/330c10e3c7a41f96ff875db1ce209871c195f4b8)
The signs of , and agree with those of their reciprocal functions , and .
![\ cot](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5bce4c1822b49c4e7ecd7b9aa07f2fcb1706817)
![\ sec](https://wikimedia.org/api/rest_v1/media/math/render/svg/3dd93d9784c55e8ef40f81993114bd3c3d01084c)
![\ csc](https://wikimedia.org/api/rest_v1/media/math/render/svg/399542cf3e9a873371132ffebe579187d48962ae)
![\ tan](https://wikimedia.org/api/rest_v1/media/math/render/svg/b57c91704d9ab0366a6436869e2968491efc5155)
![\ cos](https://wikimedia.org/api/rest_v1/media/math/render/svg/e473a3de151d75296f141f9f482fe59d582a7509)
![\ sin](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee55beec18afd710e7ab767964b915b020c65093)
Important functional values
Representation of important function values of sine and cosine on the unit circle
(°)
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Many other values can be represented.
Symmetries
The trigonometric functions have simple symmetries:
![\ sin (-x) = - \ sin x \;](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ef17e7d6bfc4e56df0f5b3fd79cb14359053e79)
![\ cos (-x) = + \ cos x \;](https://wikimedia.org/api/rest_v1/media/math/render/svg/64aaa96cfbc98220c9bb0efd88c2327149bf6e11)
![\ tan (-x) = - \ tan x \;](https://wikimedia.org/api/rest_v1/media/math/render/svg/11386acf08841533d9d77ea84602d4c961dee9bd)
![\ cot (-x) = - \ cot x \;](https://wikimedia.org/api/rest_v1/media/math/render/svg/a11bee7787b54c6ee52dd21d436faf2e701a5859)
![\ sec (-x) = + \ sec x \;](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f39c657690e30a98012cf226fa49d041ac1469a)
![\ csc (-x) = - \ csc x \;](https://wikimedia.org/api/rest_v1/media/math/render/svg/6aa0e32ab35bb996e89600b8586e7b07c87b95ce)
Phase shifts
![\ sin \ left (x + {\ frac {\ pi} {2}} \ right) = \ cos x \; \ quad {\ text {or}} \ quad \ sin \ left (x + 90 ^ {{\ circ}} \ right) = \ cos x \;](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6f15b519e7cfd7df36e90e6e7a54915ff7a4632)
![\ cos \ left (x + {\ frac {\ pi} {2}} \ right) = - \ sin x \; \ quad {\ text {or}} \ quad \ cos \ left (x + 90 ^ {{ \ circ}} \ right) = - \ sin x \;](https://wikimedia.org/api/rest_v1/media/math/render/svg/1aeef04cf4e11b3f5b6033fdbbc9966595f83faa)
![\ tan \ left (x + {\ frac {\ pi} {2}} \ right) = - \ cot x \; \ quad {\ text {or}} \ quad \ tan \ left (x + 90 ^ {{ \ circ}} \ right) = - \ cot x \;](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ffa7095211675280c890d2eb511c3da38edcdeb)
![{\ displaystyle \ cot \ left (x + {\ frac {\ pi} {2}} \ right) = - \ tan x \; \ quad {\ text {or}} \ quad \ cot \ left (x + 90 ^ {\ circ} \ right) = - \ tan x \;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8245fcdf9b7221c7ffab7f8952e793af329a1aa)
Reduction to acute angles
![{\ displaystyle \ sin x \ \; = \; \; \; \ sin \ left (\ pi -x \ right) \, \ quad {\ text {or}} \ quad \ sin x \ = \; \ ; \; \ sin \ left (180 ^ {\ circ} -x \ right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d6d53f3be5eb04c7598c7973f71f4ff7393cbcc)
![{\ displaystyle \ cos x \ \, = - \ cos \ left (\ pi -x \ right) \ quad {\ text {or}} \ quad \ cos x \ = - \ cos \ left (180 ^ {\ circ} -x \ right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2cd3b6ceeb5f97594b26c76257c83501088f28d4)
![{\ displaystyle \ tan x \ = - \ tan \ left (\ pi -x \ right) \ quad {\ text {or}} \ quad \ tan x \ = - \ tan \ left (180 ^ {\ circ} -x \ right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b73e1edccfb614e51ce9bca609dec6f801208453)
Represented by the tangent of the half angle
With the label
, the following relationships apply to anything![t = \ tan \ tfrac {x} {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef970dda45f18d891ea64d38b3b8273f0400426d)
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Addition theorems
For sine and cosine, the addition theorems can be derived from the concatenation of two rotations about the angle or . This is fundamentally possible; Reading the formulas from the product of two rotary matrices of the plane is much easier . Alternatively, the addition theorems follow from the application of Euler's formula to the relationship . The results for the double sign are obtained by applying the symmetries .
![x](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
![y](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d)
![\ mathbb {R} ^ {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd)
![{\ displaystyle \ textstyle e ^ {i (x + y)} = e ^ {ix} \ cdot e ^ {iy}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf8784c2845d219aa33fc7a84a75213ddddeb241)
![{\ displaystyle \ sin (x \ pm y) = \ sin x \ cdot \ cos y \ pm \ cos x \ cdot \ sin y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6586aa07fe77e9ccd8617dbd587397a0ba025fe2)
![{\ displaystyle \ cos (x \ pm y) = \ cos x \ cdot \ cos y \ mp \ sin x \ cdot \ sin y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a52e5b827da7c8844079ed60c12a14318599303)
By expanding with or and simplifying the double fraction:
![{\ displaystyle \ textstyle {1 \ over \ cos x \ cos y}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1db7489d2eb01eb1dc4e9f6d77666aa67c13c5bb)
![{\ displaystyle \ textstyle {1 \ over \ sin x \ sin y}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85372887a7805cd1c4693133bad7362a4b16602b)
![{\ displaystyle \ tan (x \ pm y) = {\ frac {\ sin (x \ pm y)} {\ cos (x \ pm y)}} = {\ frac {\ tan x \ pm \ tan y} {1 \ mp \ tan x \; \ tan y}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c07598e198624b2952942ddbd9ba8cdf4ff93ba)
![{\ displaystyle \ cot (x \ pm y) = {\ frac {\ cos (x \ pm y)} {\ sin (x \ pm y)}} = {\ frac {\ cot x \ cot y \ mp 1 } {\ cot y \ pm \ cot x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4ffafd223fa4565d7f72778bc8aacaa83ca7119)
For the double angle functions follow from this , for the phase shifts .
![x = y](https://wikimedia.org/api/rest_v1/media/math/render/svg/409a91214d63eabe46ec10ff3cbba689ab687366)
![y = \ pi / 2](https://wikimedia.org/api/rest_v1/media/math/render/svg/4dcd5ad6ff49e351c5c37f112a6ba624df0bf19c)
![{\ displaystyle \ sin (x + y) \ cdot \ sin (xy) = \ cos ^ {2} y- \ cos ^ {2} x = \ sin ^ {2} x- \ sin ^ {2} y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a117161ac4a44022dae917d573db51744d5f6479)
![{\ displaystyle \ cos (x + y) \ cdot \ cos (xy) = \ cos ^ {2} y- \ sin ^ {2} x = \ cos ^ {2} x- \ sin ^ {2} y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b940a0d1f7cb9846f913cc4935de4516614d83c)
Addition theorems for arc functions
The following addition theorems apply to
the arc functions
Summands
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Molecular formula
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Scope
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or
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and and![y> 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/c973e3cbfee5d7ab9ca2348b578b6ec19a8c019a)
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and and![y <0](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd6a57e5eb282c81c2bb6f5e313012fa77bc08a4)
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or
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and and![y <0](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd6a57e5eb282c81c2bb6f5e313012fa77bc08a4)
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and and![y> 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/c973e3cbfee5d7ab9ca2348b578b6ec19a8c019a)
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and
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and
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and
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and
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Double angle functions
![\ sin (2x) = 2 \ sin x \; \ cos x = \ frac {2 \ tan x} {1 + \ tan ^ 2 x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/399d9bf006f51a821e6cc5fb20eb6d6d87011731)
![\ cos (2x) = \ cos ^ 2 x - \ sin ^ 2 x = 1 - 2 \ sin ^ 2 x = 2 \ cos ^ 2 x - 1 = \ frac {1 - \ tan ^ 2 x} {1 + \ tan ^ 2 x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37a3ba745dfb5da99ca97f0b98be5ac91ef84985)
![\ tan (2x) = \ frac {2 \ tan x} {1 - \ tan ^ 2 x} = \ frac {2} {\ cot x - \ tan x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2cc333f1010df6b985309c126f59c59f5697b1c8)
![\ cot (2x) = \ frac {\ cot ^ 2 x - 1} {2 \ cot x} = \ frac {\ cot x - \ tan x} {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/39d9abccd9ffc7690cd93300763f3f3092293644)
Trigonometric functions for further multiples
The formulas for multiples are usually calculated using the complex numbers from the Euler formula
and the DeMoivre formula
. This results in
. Decomposition into real and imaginary parts then provides the formulas for
and
or the general series representation.
![{\ displaystyle z = r \ left (\ cos \ phi + i \ sin \ phi \ right) \ iff z ^ {n} = r ^ {n} \ left (\ cos \ phi + i \ sin \ phi \ right ) ^ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a8760cdba84ce61631bfc7bb7a27c071e180f30)
![{\ displaystyle z ^ {n} = r ^ {n} \ left (\ cos \ left (n \ phi \ right) + i \ sin \ left (n \ phi \ right) \ right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/437da3cf908500f01e89b270975a0b8179095159)
![{\ displaystyle \ cos \ left (n \ phi \ right) + i \ sin \ left (n \ phi \ right) = \ left (\ cos \ phi + i \ sin \ phi \ right) ^ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/513b6fb6582557faffea01cc3e63cf312b869615)
![{\ displaystyle \ cos}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e473a3de151d75296f141f9f482fe59d582a7509)
![{\ displaystyle \ sin}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee55beec18afd710e7ab767964b915b020c65093)
The formula for is related to the Chebyshev polynomials .
![\ cos (nx)](https://wikimedia.org/api/rest_v1/media/math/render/svg/a930a1025613b667060f1659b66d739d62576232)
![T_n (\ cos x) = \ cos (nx)](https://wikimedia.org/api/rest_v1/media/math/render/svg/e522492b2f2834b76fcbfbc985a7e1c6746c601f)
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![= \; \ sin x \ left (4 \ cos ^ 2 x - 1 \ right)](https://wikimedia.org/api/rest_v1/media/math/render/svg/aaa9c56168861254f7e3e9641dac72aea1f206c1)
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![= \; \ sin x \ left (8 \ cos ^ 3 x - 4 \ cos x \ right)](https://wikimedia.org/api/rest_v1/media/math/render/svg/77d4cf8c1d4f708af973d0416ba02a285239bab4)
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![= \; \ sin x \ left (16 \ cos ^ 4 x - 12 \ cos ^ 2 x + 1 \ right)](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5a04622cf8d8e7d716edf39cccf39b23e7e5b5f)
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![= \; \ sum_ {j = 0} ^ {\ lfloor \ frac {n-1} {2} \ rfloor} (-1) ^ {j} {n \ choose 2j + 1} \ sin ^ {2j + 1} x \ ; \ cos ^ {n - 2j - 1} x](https://wikimedia.org/api/rest_v1/media/math/render/svg/5508b849322fee3e7556711303c9084db4942249)
![= \; \ sin x \ sum_ {k = 0} ^ {\ lfloor \ frac {n-1} {2} \ rfloor} (-1) ^ k {nk-1 \ choose k} 2 ^ {n-2k-1} \ cos ^ {n-2k-1} x](https://wikimedia.org/api/rest_v1/media/math/render/svg/61f4c7471d54e6805ff926339b303cc16b751828)
![\ cos (3x) = 4 \ cos ^ 3 x - 3 \ cos x \,](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6adcf1cdf2f52a4dfaa35e00b09d997e97f8f7b)
![\ cos (4x) = 8 \ cos ^ 4 x - 8 \ cos ^ 2 x + 1 \,](https://wikimedia.org/api/rest_v1/media/math/render/svg/401d0c814232b3a047ee41e3e2da9e48b3acb4bd)
![\ cos (5x) = 16 \ cos ^ 5 x - 20 \ cos ^ 3 x + 5 \ cos x \,](https://wikimedia.org/api/rest_v1/media/math/render/svg/654f65892ad5763726a2d59aac3a4fa110e678f5)
![\ cos (6x) = 32 \ cos ^ 6 x - 48 \ cos ^ 4 x + 18 \ cos ^ 2 x - 1 \,](https://wikimedia.org/api/rest_v1/media/math/render/svg/43af6ba8f12d695a3e5762a79a0e709a8d3b29cb)
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![= \; \ sum_ {j = 0} ^ {\ lfloor \ frac {n} {2} \ rfloor} (-1) ^ {j} {n \ choose 2j} \ sin ^ {2j} x \; \ cos ^ {n - 2j} x](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca5ee013f0dbe44966d89b77a046d93251d1315d)
![\ tan (3x) = \ frac {3 \ tan x - \ tan ^ 3 x} {1 - 3 \ tan ^ 2 x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82fa74b1e061e40a6ec3927afa434f3ed22f18c1)
![\ tan (4x) = \ frac {4 \ tan x - 4 \ tan ^ 3 x} {1 - 6 \ tan ^ 2 x + \ tan ^ 4 x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4474fad189f0b4c965a61b9b3ffe6836659087bb)
![\ cot (3x) = \ frac {\ cot ^ 3 x - 3 \ cot x} {3 \ cot ^ 2 x - 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b61d0b1c995e3b835901bcf114e0d26f67f195e)
![\ cot (4x) = \ frac {\ cot ^ 4 x - 6 \ cot ^ 2 x + 1} {4 \ cot ^ 3 x - 4 \ cot x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5989fc85830b4459560650545d9eaae3d36ed217)
Half-angle formulas
The half- angle formulas , which can be derived from the double-angle formulas by means of substitution, are used to calculate the function value of the half argument :
![{\ displaystyle \ sin {\ frac {x} {2}} = {\ sqrt {\ frac {1- \ cos x} {2}}} \ quad {\ text {for}} \ quad x \ in \ left [0.2 \ pi \ right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c05b1bef0e3ec010ffd9aeb20f6b5fcd5facc1c9)
![{\ displaystyle \ cos {\ frac {x} {2}} = {\ sqrt {\ frac {1+ \ cos x} {2}}} \ quad {\ text {for}} \ quad x \ in \ left [- \ pi, \ pi \ right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f10157e35e43a63caa173267e34c4729ab69247)
![{\ displaystyle \ tan {\ frac {x} {2}} = {\ frac {1- \ cos x} {\ sin x}} = {\ frac {\ sin x} {1+ \ cos x}} = {\ sqrt {\ frac {1- \ cos x} {1+ \ cos x}}} \ quad {\ text {for}} \ quad x \ in \ left [0, \ pi \ right [}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10f31fd3c7ea2268fbbb2e31afe37b44157125df)
![{\ displaystyle \ cot {\ frac {x} {2}} = {\ frac {1+ \ cos x} {\ sin x}} = {\ frac {\ sin x} {1- \ cos x}} = {\ sqrt {\ frac {1+ \ cos x} {1- \ cos x}}} \ quad {\ text {for}} \ quad x \ in \ left] 0, \ pi \ right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7af43f5729ce3ece85f01d2895d1eb8013ad614)
In addition:
![{\ displaystyle \ tan {\ frac {x} {2}} = {\ frac {\ tan x} {1 + {\ sqrt {1+ \ tan ^ {2} x}}}} \ quad {\ text { for}} \ quad x \ in \ left] - {\ tfrac {\ pi} {2}}, {\ tfrac {\ pi} {2}} \ right [}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b5e598d507ce768e3efd5bfb8bf974e60b004bb)
![{\ displaystyle \ cot {\ frac {x} {2}} = \ cot x + {\ sqrt {1+ \ cot ^ {2} x}} \ quad {\ text {for}} \ quad x \ in \ left ] 0, \ pi \ right [}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab72611c37657b6aedab9984b9976053a5213dab)
See also: half-angle block
Sums of two trigonometric functions (identities)
Identities can be derived from the addition theorems , with the help of which the sum of two trigonometric functions can be represented as a product:
![\ sin x + \ sin y = 2 \ sin \ frac {x + y} {2} \ cos \ frac {xy} {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e01cf1e797f9464e17c37cfcc5dde3c5f13912e5)
![\ sin x- \ sin y = 2 \ cos \ frac {x + y} {2} \ sin \ frac {xy} {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0adc6913782f902aa3caad9ff414c66add66aa94)
![\ cos x + \ cos y = 2 \ cos \ frac {x + y} {2} \ cos \ frac {xy} {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2791d88f49dafee372535fdaed0480528162041d)
![\ cos x- \ cos y = -2 \ sin {\ frac {x + y} {2}} \ sin {\ frac {xy} {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c581a11aaaa29367b9f2dc636effd710aecb2e07)
![\ left. \ begin {matrix} \ tan x + \ tan y = \ dfrac {\ sin (x + y)} {\ cos x \ cos y} \\ [1em] \ tan x- \ tan y = \ dfrac {\ sin ( xy)} {\ cos x \ cos y} \ end {matrix} \ right \} \ Rightarrow \ tan x \ pm \ tan y = \ frac {\ sin (x \ pm y)} {\ cos x \ cos y }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c82f5271400544871570c9e8e755362c0c27390c)
![\ left. \ begin {matrix} \ cot x + \ cot y = \ dfrac {\ sin (y + x)} {\ sin x \ sin y} \\ [1em] \ cot x - \ cot y = \ dfrac {\ sin (yx)} {\ sin x \ sin y} \ end {matrix} \ right \} \ Rightarrow \ cot x \ pm \ cot y = \ frac {\ sin (y \ pm x)} {\ sin x \ sin y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/324f139ea94aaeceb07ee500aa35eaa16d0755c8)
This results in special cases:
![\ cos x + \ sin x = \ sqrt {2} \ cdot \ sin \ left (x + \ frac {\ pi} {4} \ right) = \ sqrt {2} \ cdot \ cos \ left (x - \ frac {\ pi} {4} \ right)](https://wikimedia.org/api/rest_v1/media/math/render/svg/66c8a83d0f2edb868b192b3e474eea5a76853ad1)
![\ cos x - \ sin x = \ sqrt {2} \ cdot \ cos \ left (x + \ frac {\ pi} {4} \ right) = - \ sqrt {2} \ cdot \ sin \ left (x - \ frac {\ pi} {4} \ right)](https://wikimedia.org/api/rest_v1/media/math/render/svg/faf14e5c9269855353fb61d1f8e4e25abcb6a8f0)
Products of the trigonometric functions
Products of the trigonometric functions can be calculated using the following formulas:
![\ sin x \; \ sin y = \ frac {1} {2} \ Big (\ cos (xy) - \ cos (x + y) \ Big)](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbdf85ce68f09d28cd51e5b2b38099e44f798f1b)
![\ cos x \; \ cos y = \ frac {1} {2} \ Big (\ cos (xy) + \ cos (x + y) \ Big)](https://wikimedia.org/api/rest_v1/media/math/render/svg/b245641b8e0f75516322c49bf3edad0434a206cc)
![\ sin x \; \ cos y = \ frac {1} {2} \ Big (\ sin (xy) + \ sin (x + y) \ Big)](https://wikimedia.org/api/rest_v1/media/math/render/svg/a70353bdec37afab686c482cbdffb2cfe0f8f10f)
![\ tan x \; \ tan y = \ frac {\ tan x + \ tan y} {\ cot x + \ cot y} = - \ frac {\ tan x - \ tan y} {\ cot x - \ cot y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f264236ebb81fc31a499d5ecf60be754ac05c9a)
![\ cot x \; \ cot y = \ frac {\ cot x + \ cot y} {\ tan x + \ tan y} = - \ frac {\ cot x - \ cot y} {\ tan x - \ tan y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27c57817c66ced9696131d1a88600c338664fd46)
![\ tan x \; \ cot y = \ frac {\ tan x + \ cot y} {\ cot x + \ tan y} = - \ frac {\ tan x - \ cot y} {\ cot x - \ tan y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f71971336fa0490cec07645e42bbdad9d907766a)
![\ sin x \; \ sin y \; \ sin z = \ frac {1} {4} \ Big (\ sin (x + yz) + \ sin (y + zx) + \ sin (z + xy) - \ sin (x + y + z) \ Big) )](https://wikimedia.org/api/rest_v1/media/math/render/svg/fdfa4793e98f7471e76ce0d9cfcf5e2f3c7a61c2)
![\ cos x \; \ cos y \; \ cos z = \ frac {1} {4} \ Big (\ cos (x + yz) + \ cos (y + zx) + \ cos (z + xy) + \ cos (x + y + z) \ Big) )](https://wikimedia.org/api/rest_v1/media/math/render/svg/a931948e89d11c857f10d40e2477490d0aeb0006)
![\ sin x \; \ sin y \; \ cos z = \ frac {1} {4} \ Big (- \ cos (x + yz) + \ cos (y + zx) + \ cos (z + xy) - \ cos (x + y + z) \ Big)](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0762f0e9b8a98c0adc77431f32fa639b44d723b)
![\ sin x \; \ cos y \; \ cos z = \ frac {1} {4} \ Big (\ sin (x + yz) - \ sin (y + zx) + \ sin (z + xy) + \ sin (x + y + z) \ Big) )](https://wikimedia.org/api/rest_v1/media/math/render/svg/668ada9f6d993b5369975ec4444c24d65f185e56)
From the double angle function for it also follows:
![\ sin (2x)](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e2822c6491b347058bf997b431f1a87ff412a44)
![\ sin x \; \ cos x = \ frac {1} {2} \ sin (2x)](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e7b7c92f4973ad7ac2a6de20189a496f96d472b)
Powers of the trigonometric functions
Sine
![\ sin ^ 2 x = \ frac {1} {2} \ \ Big (1 - \ cos (2x) \ Big)](https://wikimedia.org/api/rest_v1/media/math/render/svg/5fe1eb0db7c2d60fb7f8d740d2803e88c18ade4c)
![{\ displaystyle \ sin ^ {3} x = {\ frac {1} {4}} \ {\ Big (} 3 \, \ sin x- \ sin (3x) {\ Big)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03888a7e5d05a4fa1a55388cb4cbebbcffcc8060)
![{\ displaystyle \ sin ^ {4} x = {\ frac {1} {8}} \ {\ Big (} 3-4 \, \ cos (2x) + \ cos (4x) {\ Big)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0486b0d3f608dbd4f8e8175841a3893fc1b8fe29)
![{\ displaystyle \ sin ^ {5} x = {\ frac {1} {16}} \ {\ Big (} 10 \, \ sin x-5 \, \ sin (3x) + \ sin (5x) {\ Big)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e3c742bb425a373df188ec22f447ad8fb88bd59)
![{\ displaystyle \ sin ^ {6} x = {\ frac {1} {32}} \ {\ Big (} 10-15 \, \ cos (2x) +6 \, \ cos (4x) - \ cos ( 6x) {\ Big)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8c19b66c5e0a5862fb44a4746ad5cf4208ea5ed)
![{\ displaystyle \ sin ^ {n} x = {\ frac {1} {2 ^ {n}}} \, \ sum _ {k = 0} ^ {n} {n \ choose k} \, \ cos { \ Big (} (n-2k) (x - {\ frac {\ pi} {2}} \) {\ Big)} \; \ quad n \ in \ mathbb {N}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c33ed56ba1f8a273aed865b11e87058405cc909d)
![{\ displaystyle \ sin ^ {n} x = {\ frac {1} {2 ^ {n}}} \, {n \ choose {\ frac {n} {2}}} + {\ frac {1} { 2 ^ {n-1}}} \ sum _ {k = 0} ^ {{\ frac {n} {2}} - 1} (- 1) ^ {{\ frac {n} {2}} - k } \, {n \ choose k} \, \ cos {((n-2k) x)}; \ quad n \ in \ mathbb {N} {\ text {and}} n {\ text {even}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9563ad90f6a8943e47efd46e3655fbe60d8d9d6)
![{\ displaystyle \ sin ^ {n} x = {\ frac {1} {2 ^ {n-1}}} \, \ sum _ {k = 0} ^ {\ frac {n-1} {2}} (-1) ^ {{\ frac {n-1} {2}} - k} \, {n \ choose k} \, \ sin {((n-2k) x)}; \ quad n \ in \ mathbb {N} {\ text {and}} n {\ text {odd}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0eb3328288183e462d3665259431d79446d3b443)
cosine
![\ cos ^ 2 x = \ frac {1} {2} \ \ Big (1 + \ cos (2x) \ Big)](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ce1bce60e3ce89b72d63d9520c1c6fdfb01a0ba)
![{\ displaystyle \ cos ^ {3} x = {\ frac {1} {4}} \ {\ Big (} 3 \, \ cos x + \ cos (3x) {\ Big)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/61dce131c936a103d315f71180f93d3d034d300a)
![{\ displaystyle \ cos ^ {4} x = {\ frac {1} {8}} \ {\ Big (} 3 + 4 \, \ cos (2x) + \ cos (4x) {\ Big)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2acba9d1ccf7ab25b3c464ee2435db78affdc7a)
![{\ displaystyle \ cos ^ {5} x = {\ frac {1} {16}} \ {\ Big (} 10 \, \ cos x + 5 \, \ cos (3x) + \ cos (5x) {\ Big)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bad0eff417901c19314dfd5281c6a2e67fc7ea99)
![{\ displaystyle \ cos ^ {6} x = {\ frac {1} {32}} \ {\ Big (} 10 + 15 \, \ cos (2x) +6 \, \ cos (4x) + \ cos ( 6x) {\ Big)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0e2f94aa463a71f88ee8ee4f982a5d935553bfc)
![{\ displaystyle \ cos ^ {n} x = {\ frac {1} {2 ^ {n}}} \, \ sum _ {k = 0} ^ {n} {n \ choose k} \, \ cos ( (n-2k) x); \ quad n \ in \ mathbb {N}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0747fe412464be2dd2f102c1f935963b7385c474)
![{\ displaystyle \ cos ^ {n} x = {\ frac {1} {2 ^ {n}}} \, {n \ choose {\ frac {n} {2}}} + {\ frac {1} { 2 ^ {n-1}}} \ sum _ {k = 0} ^ {{\ frac {n} {2}} - 1} {n \ choose k} \, \ cos {((n-2k) x )}; \ quad n \ in \ mathbb {N} {\ text {and}} n {\ text {even}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/948ee4681c13e1a87c46510f1056924428ab2322)
![{\ displaystyle \ cos ^ {n} x = {\ frac {1} {2 ^ {n-1}}} \, \ sum _ {k = 0} ^ {\ frac {n-1} {2}} {n \ choose k} \, \ cos {((n-2k) x)}; \ quad n \ in \ mathbb {N} {\ text {and}} n {\ text {odd}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f09f6e84cb62249d3c20139fb902c6c3508a89c0)
tangent
![{\ displaystyle \ tan ^ {2} x = {\ frac {1- \ cos (2x)} {1+ \ cos (2x)}} = \ sec ^ {2} (x) -1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f240daafd1c60088ccd30555950244e55e5365a)
Conversion into other trigonometric functions
![\ sin (\ arccos x) = \ cos (\ arcsin x) = {\ sqrt {1-x ^ {2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb3d925c7516a151e19c1214aeb9b27c83447d44)
![\ sin (\ arctan x) = \ cos (\ operatorname {arccot} x) = {\ frac {x} {{\ sqrt {1 + x ^ {2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/390c2be0f9a72cf04719626c37c60259e63dd0e8)
![\ sin (\ operatorname {arccot} x) = \ cos (\ arctan x) = {\ frac {1} {{\ sqrt {1 + x ^ {2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7b0f80a248c79c1bcb02755a611e206c2a723b7)
![\ tan (\ arcsin x) = \ cot (\ arccos x) = {\ frac {x} {{\ sqrt {1-x ^ {2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c56c5d7ce7b22c46deb6223478d8a21195d3326)
![\ tan (\ arccos x) = \ cot (\ arcsin x) = {\ frac {{\ sqrt {1-x ^ {2}}}} {x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24299451f1275ac355d38f6b76d6b7a0d5a0f121)
![\ tan (\ operatorname {arccot} x) = \ cot (\ arctan x) = {\ frac {1} {x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9332541067936f15f11b173aefc580b162394618)
Further formulas for the case α + β + γ = 180 °
The following formulas apply to any plane triangles and follow after long term transformations , as long as the functions in the formulas are well-defined (the latter only applies to formulas in which the tangent and cotangent occur).
![\ alpha + \ beta + \ gamma = 180 ^ {\ circ}](https://wikimedia.org/api/rest_v1/media/math/render/svg/720f59b296cd42f5d2968b211dd56394a860c97a)
![\ tan \ alpha + \ tan \ beta + \ tan \ gamma = \ tan \ alpha \ cdot \ tan \ beta \ cdot \ tan \ gamma \,](https://wikimedia.org/api/rest_v1/media/math/render/svg/0fe8c152093098d464f496dd0de3913f1529ca86)
![\ cot \ beta \ cdot \ cot \ gamma + \ cot \ gamma \ cdot \ cot \ alpha + \ cot \ alpha \ cdot \ cot \ beta = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/74fcc3c197ed624c70b6cfc00746cb2ef548c046)
![{\ displaystyle \ cot {\ frac {\ alpha} {2}} + \ cot {\ frac {\ beta} {2}} + \ cot {\ frac {\ gamma} {2}} = \ cot {\ frac {\ alpha} {2}} \ cdot \ cot {\ frac {\ beta} {2}} \ cdot \ cot {\ frac {\ gamma} {2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a62b0958247ab0e8f1bc729f2678941a1addedc1)
![\ tan \ frac {\ beta} {2} \ tan \ frac {\ gamma} {2} + \ tan \ frac {\ gamma} {2} \ tan \ frac {\ alpha} {2} + \ tan \ frac {\ alpha} {2} \ tan \ frac {\ beta} {2} = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/aca762a805db0c80b84d38043a0de29a702f45f1)
![\ sin \ alpha + \ sin \ beta + \ sin \ gamma = 4 \ cos \ frac {\ alpha} {2} \ cos \ frac {\ beta} {2} \ cos \ frac {\ gamma} {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78731cad789eccabfe272fc5d2a85dbd1c044da6)
![- \ sin \ alpha + \ sin \ beta + \ sin \ gamma = 4 \ cos \ frac {\ alpha} {2} \ sin \ frac {\ beta} {2} \ sin \ frac {\ gamma} {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/512cb66b272c60f4339f2d55f794701a44fe3e6c)
![\ cos \ alpha + \ cos \ beta + \ cos \ gamma = 4 \ sin \ frac {\ alpha} {2} \ sin \ frac {\ beta} {2} \ sin \ frac {\ gamma} {2} + 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a518b4be27361bd289667bee5e7305ba8789336)
![- \ cos \ alpha + \ cos \ beta + \ cos \ gamma = 4 \ sin \ frac {\ alpha} {2} \ cos \ frac {\ beta} {2} \ cos \ frac {\ gamma} {2} -1](https://wikimedia.org/api/rest_v1/media/math/render/svg/43a71ae15010ea24ecb46d8e389a1b48b4a232a3)
![{\ displaystyle \ sin (2 \ alpha) + \ sin (2 \ beta) + \ sin (2 \ gamma) = 4 \ sin \ alpha \ sin \ beta \ sin \ gamma}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84b2ab82cbb3ac0a4ba782ddc919af3d3390444d)
![{\ displaystyle - \ sin (2 \ alpha) + \ sin (2 \ beta) + \ sin (2 \ gamma) = 4 \ sin \ alpha \ cos \ beta \ cos \ gamma}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebd06ed342819fd52aa48ae8e3b8fb856cfc9ccc)
![{\ displaystyle \ cos (2 \ alpha) + \ cos (2 \ beta) + \ cos (2 \ gamma) = - 4 \ cos \ alpha \ cos \ beta \ cos \ gamma -1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ec8f429afcc012859e67209a52262994283a755)
![{\ displaystyle - \ cos (2 \ alpha) + \ cos (2 \ beta) + \ cos (2 \ gamma) = - 4 \ cos \ alpha \ sin \ beta \ sin \ gamma +1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/960452152b02d398725c66d1dde4676c6e5d1bb6)
![{\ displaystyle \ sin ^ {2} \ alpha + \ sin ^ {2} \ beta + \ sin ^ {2} \ gamma = 2 \ cos \ alpha \ cos \ beta \ cos \ gamma +2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf13c2d74290030db3a16884cd09d405dd514ac3)
![{\ displaystyle - \ sin ^ {2} \ alpha + \ sin ^ {2} \ beta + \ sin ^ {2} \ gamma = 2 \ cos \ alpha \ sin \ beta \ sin \ gamma}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e589036b21cf22bf9ba257966e22626d3da37dee)
![{\ displaystyle \ cos ^ {2} \ alpha + \ cos ^ {2} \ beta + \ cos ^ {2} \ gamma = -2 \ cos \ alpha \ cos \ beta \ cos \ gamma +1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe5b5a75977258c9b11ad3f75fb710625a426a29)
![{\ displaystyle - \ cos ^ {2} \ alpha + \ cos ^ {2} \ beta + \ cos ^ {2} \ gamma = -2 \ cos \ alpha \ sin \ beta \ sin \ gamma +1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/045c51d9667bdb6836408b9b49d8e4e30abdf104)
![{\ displaystyle - \ sin ^ {2} (2 \ alpha) + \ sin ^ {2} (2 \ beta) + \ sin ^ {2} (2 \ gamma) = - 2 \ cos (2 \ alpha) \ , \ sin (2 \ beta) \, \ sin (2 \ gamma)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3376d32e16f1f5b0f47b32da2dd19d90e10eea30)
![- \ cos ^ {2} (2 \ alpha) + \ cos ^ {2} (2 \ beta) + \ cos ^ {2} (2 \ gamma) = 2 \ cos (2 \ alpha) \, \ sin ( 2 \ beta) \, \ sin (2 \ gamma) +1](https://wikimedia.org/api/rest_v1/media/math/render/svg/93ebcefd3c91506fcd127a539695aa98fbd41b60)
![\ sin ^ {{2}} \ left ({\ frac {\ alpha} {2}} \ right) + \ sin ^ {{2}} \ left ({\ frac {\ beta} {2}} \ right ) + \ sin ^ {{2}} \ left ({\ frac {\ gamma} {2}} \ right) +2 \ sin \ left ({\ frac {\ alpha} {2}} \ right) \, \ sin \ left ({\ frac {\ beta} {2}} \ right) \, \ sin \ left ({\ frac {\ gamma} {2}} \ right) = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/8df190336bc9c4957d82852c345f3e3b503bb5bd)
Sinusoid and linear combination with the same phase
![\ begin {align} a \ sin \ alpha + b \ cos \ alpha = & \ begin {cases} \ sqrt {a ^ 2 + b ^ 2} \ sin \ left (\ alpha + \ arctan \ left (\ tfrac { b} {a} \ right) \ right) & \ text {, for all} a> 0 \\ \ sqrt {a ^ 2 + b ^ 2} \ cos \ left (\ alpha - \ arctan \ left (\ tfrac {a} {b} \ right) \ right) & \ text {, for all} b> 0 \ end {cases} \ end {align}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08b4b9d1ee876c5d8a75fcd2807e7be8a4501c93)
![{\ displaystyle {\ begin {aligned} a \ cos \ alpha + b \ sin \ alpha = \ operatorname {sgn} (a) {\ sqrt {a ^ {2} + b ^ {2}}} \ cos \ left (\ alpha + \ arctan \ left (- {\ tfrac {b} {a}} \ right) \ right) \ end {aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74ce17383bb44184817be083a504ec7632af4189)
![a \ sin (x + \ alpha) + b \ sin (x + \ beta) = \ sqrt {a ^ 2 + b ^ 2 + 2a b \ cos (\ alpha- \ beta)} \ cdot \ sin (x + \ delta) ,](https://wikimedia.org/api/rest_v1/media/math/render/svg/a366244bdf45de7731ac858703b8350e627c247d)
in which
Is more general
![\ sum_i a_i \ sin (x + \ delta_i) = a \ sin (x + \ delta),](https://wikimedia.org/api/rest_v1/media/math/render/svg/6436361f9869baa8e2dfb11f06aa19247d9e087a)
in which
![a ^ 2 = \ sum_ {i, j} a_i a_j \ cos (\ delta_i- \ delta_j)](https://wikimedia.org/api/rest_v1/media/math/render/svg/17d757c684ae3fdce8dca0b3b1b9bdf44555cf11)
and
![\ delta = \ operatorname {atan2} \ left (\ sum_i a_i \ sin \ delta_i, \ sum_i a_i \ cos \ delta_i \ right).](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ca1f584829a4adb251629bc8b258f1edf5191ce)
Derivatives and antiderivatives
See the formula collection derivatives and antiderivatives
Series development
The sine (red) compared to its 7th Taylor polynomial (green)
As in other calculus , all angles are given in radians .
It can be shown that the cosine is the derivative of the sine and the derivative of the cosine is the negative sine. Having these derivatives, one can expand the Taylor series (easiest with the expansion point ) and show that the following identities hold for all of the real numbers . These series are used to define the trigonometric functions for complex arguments ( or denote the Bernoulli numbers ):
![x = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc)
![x](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
![B_ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f568bf6d34e97b9fdda0dc7e276d6c4501d2045)
![\ beta_n](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f6cd4f61e83a945551749f7384429eeee37b2b1)
![\ begin {align} \ sin x & = \ sum_ {n = 0} ^ \ infty \ frac {(- 1) ^ n} {(2n + 1)!} x ^ {2n + 1} \\ & = x - \ frac {x ^ 3} {3!} + \ frac {x ^ 5} {5!} - \ frac {x ^ 7} {7!} \ pm \ cdots \;, \ qquad | x | <\ infty \ end {align}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09c6a6662292ba76906959205400b479fb862116)
![\ begin {align} \ cos x & = \ sum_ {n = 0} ^ \ infty \ frac {(- 1) ^ n} {(2n)!} x ^ {2n} \\ & = 1 - \ frac { x ^ 2} {2!} + \ frac {x ^ 4} {4!} - \ frac {x ^ 6} {6!} \ pm \ cdots \;, \ qquad | x | <\ infty \ end {align}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5dfc7edf91f13dc125faf05d17849b55e40a0cab)
![{\ displaystyle {\ begin {aligned} \ tan x & = \ sum _ {n = 1} ^ {\ infty} (- 1) ^ {n} {\ frac {2 ^ {2n} (1-2 ^ {2n }) \ beta _ {2n}} {(2n)!}} x ^ {2n-1} = \ sum _ {n = 1} ^ {\ infty} {\ frac {2 ^ {2n} (2 ^ { 2n} -1) B_ {n}} {(2n)!}} X ^ {2n-1} \\ & = x + {\ frac {1} {3}} x ^ {3} + {\ frac {2 } {15}} x ^ {5} + {\ frac {17} {315}} x ^ {7} + {\ frac {62} {2835}} x ^ {9} + \, \ cdots \ qquad | x | <{\ tfrac {\ pi} {2}} \ end {aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d848aff3bb29a9d1ba966340cfabd062df4b9827)
![{\ displaystyle {\ begin {aligned} \ cot x & = {\ frac {1} {x}} - \ sum _ {n = 1} ^ {\ infty} {\ frac {\ left (-1 \ right) ^ {n-1} 2 ^ {2n} \ beta _ {2n}} {(2n)!}} x ^ {2n-1} = {\ frac {1} {x}} - \ sum _ {n = 1 } ^ {\ infty} {\ frac {2 ^ {2n} B_ {n}} {(2n)!}} x ^ {2n-1} \\ & = {\ frac {1} {x}} - { \ frac {1} {3}} x - {\ frac {1} {45}} x ^ {3} - {\ frac {2} {945}} x ^ {5} - {\ frac {1} { 4725}} x ^ {7} - \, \ cdots, \ qquad 0 <| x | <\ pi \ end {aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c855095135bb48aac17cbb012f6642ce2c9d68c0)
Product development
![\ sin (x) = x \ prod_ {k = 1} ^ \ infty \ left (1 - \ frac {x ^ 2} {k ^ 2 \ pi ^ 2} \ right)](https://wikimedia.org/api/rest_v1/media/math/render/svg/babb7ccd3c5a73b532fde2dd81644275d9178892)
![\ cos (x) = \ prod_ {k = 1} ^ \ infty \ left (1 - \ frac {4x ^ 2} {(2k-1) ^ 2 \ pi ^ 2} \ right)](https://wikimedia.org/api/rest_v1/media/math/render/svg/eee41de17dcf51a45517a0359c375d0f82fd7bd5)
![{\ displaystyle \ sin (x) = \ prod _ {n = - \ infty} ^ {\ infty} \ left ({\ frac {x + n \ pi} {{\ frac {\ pi} {2}} + n \ pi}} \ right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c6114aef9bc6d486b907f4321a1fb7bdb062657)
![{\ displaystyle \ cos (x) = \ prod _ {n = - \ infty} ^ {\ infty} \ left ({\ frac {x + n \ pi + {\ frac {\ pi} {2}}} { {\ frac {\ pi} {2}} + n \ pi}} \ right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf11993c9768e45c7a5b3d343e2cce10ebfd71df)
![{\ displaystyle \ tan (x) = \ prod _ {n = - \ infty} ^ {\ infty} \ left ({\ frac {x + n \ pi} {x + n \ pi + {\ frac {\ pi } {2}}}} \ right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92dc105991696a839dc00fe30c8222607c460edc)
![{\ displaystyle \ csc (x) = \ prod _ {n = - \ infty} ^ {\ infty} \ left ({\ frac {{\ frac {\ pi} {2}} + n \ pi} {x + n \ pi}} \ right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38cf26717eb22cdab9f9899915c15cd07115ef4d)
![{\ displaystyle \ sec (x) = \ prod _ {n = - \ infty} ^ {\ infty} \ left ({\ frac {{\ frac {\ pi} {2}} + n \ pi} {x + n \ pi + {\ frac {\ pi} {2}}}} \ right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/874ae90fbdbb1ae04d1f2e1e585ab41909d3e744)
![{\ displaystyle \ cot (x) = \ prod _ {n = - \ infty} ^ {\ infty} \ left ({\ frac {x + n \ pi + {\ frac {\ pi} {2}}} { x + n \ pi}} \ right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46ef5780d32ca536260b624f8446c98e4934fda6)
Connection with the complex exponential function
Further, there is between the functions , and the complex exponential function of the following relationship:
![\ sin x](https://wikimedia.org/api/rest_v1/media/math/render/svg/09b4b55580d6a821a07ad9fe35be88976917b10b)
![{\ displaystyle \ exp (\ mathrm {i} x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f645a50f11e7ee97b32b54095c2d52df2ed1a4f)
-
( Euler's formula )
There is still writing.
![{\ displaystyle \ cos {x} + \ mathrm {i} \ sin {x} =: \ operatorname {cis} (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e28af42ed9007d7d96e8fd7d0bc88354d3841730)
Due to the symmetries mentioned above, the following also applies:
![{\ displaystyle \ cos x = {\ frac {\ exp (\ mathrm {i} x) + \ exp (- \ mathrm {i} x)} {2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0313e645fccb1e14ea9e3d5bb148f9c24ea29cb)
![{\ displaystyle \ sin x = {\ frac {\ exp (\ mathrm {i} x) - \ exp (- \ mathrm {i} x)} {2 \ mathrm {i}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1cd04620f6a14015c6e0b38b6c9e50a918d8364c)
With these relationships, some addition theorems can be derived particularly easily and elegantly.
Spherical trigonometry
A collection of formulas for the right-angled and the general triangle on the spherical surface can be found in a separate chapter.
Literature, web links
Individual evidence
-
↑ Die Wurzel 2006/04 + 05, 104ff., Without proof
-
↑ Joachim Mohr: Cosine, sine and tangent values , accessed on June 1, 2016
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^ A b Otto Forster: Analysis 1. Differential and integral calculus of a variable. vieweg 1983, p. 87.
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^ IN Bronstein, KA Semendjajew: Taschenbuch der Mathematik . 19th edition, 1979. BG Teubner Verlagsgesellschaft, Leipzig. P. 237.
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↑ Milton Abramowitz and Irene A. Stegun, March 22 , 2015 , (see above "Web Links ")
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↑ Milton Abramowitz and Irene A. Stegun, 4.3.27 , (see also above "Weblinks")
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↑ Milton Abramowitz and Irene A. Stegun, 4.3.29 , (see above "Weblinks")
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^ IS Gradshteyn and IM Ryzhik , Table of Integrals, Series, and Products , Academic Press, 5th edition (1994). ISBN 0-12-294755-X 1.333.4
-
↑ IS Gradshteyn and IM Ryzhik, ibid 1.331.3 (In this formula, however, Gradshteyn / Ryzhik contains a sign error )
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↑ a b c d e f g h i j k l m n o I. N. Bronstein, KA Semendjajew, Taschenbuch der Mathematik , BG Teubner Verlagsgesellschaft Leipzig. 19th edition 1979. 2.5.2.1.3
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↑ Milton Abramowitz and Irene A. Stegun, 4.3.28 , (see above "Weblinks")
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↑ Milton Abramowitz and Irene A. Stegun, March 4th , 30th , (see above "Weblinks")
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↑ IS Gradshteyn and IM Ryzhik, ibid 1.335.4
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↑ IS Gradshteyn and IM Ryzhik, ibid 1.335.5
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↑ IS Gradshteyn and IM Ryzhik, ibid 1.331.3
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↑ IS Gradshteyn and IM Ryzhik, ibid 1.321.1
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↑ IS Gradshteyn and IM Ryzhik, ibid 1.321.2
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↑ IS Gradshteyn and IM Ryzhik, ibid 1.321.3
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↑ IS Gradshteyn and IM Ryzhik, ibid 1.321.4
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↑ IS Gradshteyn and IM Ryzhik, ibid 1.321.5
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↑ IS Gradshteyn and IM Ryzhik, ibid 1.323.1
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↑ IS Gradshteyn and IM Ryzhik, ibid 1.323.2
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↑ IS Gradshteyn and IM Ryzhik, ibid 1.323.3
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↑ IS Gradshteyn and IM Ryzhik, ibid 1.323.4
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↑ IS Gradshteyn and IM Ryzhik, ibid 1.323.5
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^ Weisstein, Eric W .: Harmonic Addition Theorem. Retrieved January 20, 2018 .
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↑ Milton Abramowitz and Irene A. Stegun, March 4th , 67 , (see above "Weblinks")
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↑ Milton Abramowitz and Irene A. Stegun, 4.3.70 , (see above "Weblinks")
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^ Herbert Amann, Joachim Escher: Analysis I, Birkhäuser Verlag, Basel 2006, 3rd edition, pp. 292 and 298