In the trigonometry , the law of tangents (also tangent set and control of Napier ) a relationship between the three sides of a plane triangle and the tangent of half the sum and half the difference of two angles of the triangle forth.
For the three sides a , b and c of a triangle as well as for the angles α, β and γ lying opposite these sides:
b
+
c
b
-
c
=
tan
β
+
γ
2
tan
β
-
γ
2
{\ displaystyle {\ frac {b + c} {bc}} = {\ frac {\ tan {\ frac {\ beta + \ gamma} {2}}} {\ tan {\ frac {\ beta - \ gamma} {2}}}}}
Because of this, this formula can also be written as
tan
β
+
γ
2
=
tan
180
∘
-
α
2
=
tan
(
90
∘
-
α
2
)
=
cot
α
2
{\ displaystyle \ tan {\ frac {\ beta + \ gamma} {2}} = \ tan {\ frac {180 ^ {\ circ} - \ alpha} {2}} = \ tan \ left (90 ^ {\ circ} - {\ frac {\ alpha} {2}} \ right) = \ cot {\ frac {\ alpha} {2}}}
b
+
c
b
-
c
=
cot
α
2
tan
β
-
γ
2
{\ displaystyle {\ frac {b + c} {bc}} = {\ frac {\ cot {\ frac {\ alpha} {2}}} {\ tan {\ frac {\ beta - \ gamma} {2} }}}}
Analogous formulas for and are obtained through cyclic exchange :
a
+
b
a
-
b
{\ displaystyle {\ frac {a + b} {ab}}}
a
+
c
a
-
c
{\ displaystyle {\ frac {a + c} {ac}}}
a
+
b
a
-
b
=
tan
α
+
β
2
tan
α
-
β
2
=
cot
γ
2
tan
α
-
β
2
{\ displaystyle {\ 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}}}}}
c
+
a
c
-
a
=
tan
γ
+
α
2
tan
γ
-
α
2
=
cot
β
2
tan
γ
-
α
2
{\ 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}}}}}
Because of this , one of these formulas remains valid if both the sides and the associated angles are swapped, for example:
tan
(
-
x
)
=
-
tan
(
x
)
{\ displaystyle \ tan (-x) = - \ tan (x)}
a
+
c
a
-
c
=
tan
α
+
γ
2
tan
α
-
γ
2
=
cot
β
2
tan
α
-
γ
2
{\ displaystyle {\ 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}}}}}
Proof with the law of sines and identities of the trigonometric functions
According to the law of sines , it follows that:
b
c
=
sin
β
sin
γ
{\ displaystyle {\ tfrac {b} {c}} = {\ tfrac {\ sin \ beta} {\ sin \ gamma}}}
b
+
c
b
-
c
=
{\ displaystyle {\ frac {b + c} {bc}} =}
b
c
+
1
b
c
-
1
=
{\ displaystyle {\ frac {{\ frac {b} {c}} + 1} {{\ frac {b} {c}} - 1}} =}
sin
β
sin
γ
+
sin
γ
sin
γ
sin
β
sin
γ
-
sin
γ
sin
γ
=
{\ displaystyle {\ frac {{\ frac {\ sin \ beta} {\ sin \ gamma}} + {\ frac {\ sin \ gamma} {\ sin \ gamma}}} {{\ frac {\ sin \ beta } {\ sin \ gamma}} - {\ frac {\ sin \ gamma} {\ sin \ gamma}}}} =}
sin
β
+
sin
γ
sin
β
-
sin
γ
,
{\ displaystyle {\ frac {\ sin \ beta + \ sin \ gamma} {\ sin \ beta - \ sin \ gamma}},}
after inserting the identities
sin
β
+
sin
γ
=
2
sin
β
+
γ
2
cos
β
-
γ
2
.
{\ displaystyle \ sin \ beta + \ sin \ gamma = 2 \ sin {\ frac {\ beta + \ gamma} {2}} \ cos {\ frac {\ beta - \ gamma} {2}}.}
such as
sin
β
-
sin
γ
=
2
cos
β
+
γ
2
sin
β
-
γ
2
{\ displaystyle \ sin \ beta - \ sin \ gamma = 2 \ cos {\ frac {\ beta + \ gamma} {2}} \ sin {\ frac {\ beta - \ gamma} {2}}}
,
which can be derived from the addition theorems , the desired formula is obtained by division.
Proof with Mollweid's formulas
With the sum of the angles in the triangle and transition to the complementary angle :
tan
β
+
γ
2
=
tan
180
∘
-
α
2
=
tan
(
90
∘
-
α
2
)
=
cot
α
2
;
{\ displaystyle \ tan {\ beta + \ gamma \ over 2} = \ tan {180 ^ {\ circ} - \ alpha \ over 2} = \ tan \ left (90 ^ {\ circ} - {\ alpha \ over 2} \ right) = \ cot {\ alpha \ over 2}; \ quad}
(1)
From Mollweid's formulas it follows with (1):
b
+
c
b
-
c
=
b
+
c
a
⋅
a
b
-
c
=
{\ displaystyle {b + c \ over bc} = \ quad {b + c \ over a} \ cdot {a \ over bc} = \ quad}
cos
β
-
γ
2
sin
α
2
⋅
cos
α
2
sin
β
-
γ
2
=
{\ displaystyle {\ frac {\ cos {\ frac {\ beta - \ gamma} {2}}} {\ sin {\ frac {\ alpha} {2}}}} \ cdot {\ frac {\ cos {\ frac {\ alpha} {2}}} {\ sin {\ frac {\ beta - \ gamma} {2}}}} = \ quad}
cot
β
-
γ
2
⋅
cot
α
2
=
{\ displaystyle \ cot {\ frac {\ beta - \ gamma} {2}} \ cdot \ cot {\ frac {\ alpha} {2}} = \ quad}
cot
α
2
tan
β
-
γ
2
=
{\ displaystyle {\ frac {\ cot {\ frac {\ alpha} {2}}} {\ tan {\ frac {\ beta - \ gamma} {2}}}} = \ quad}
tan
β
+
γ
2
tan
β
-
γ
2
,
{\ displaystyle {\ frac {\ tan {\ frac {\ beta + \ gamma} {2}}} {\ tan {\ frac {\ beta - \ gamma} {2}}}}, \ quad}
qed
See also
literature
Web links
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