# Tangent and cotangent

Graph of the tangent function (argument x in radians )
Diagram of the cotangent function (argument x in radians)

Tangent and cotangent are trigonometric functions and play a prominent role in mathematics and its areas of application. The tangent of the angle is denoted by, the cotangent of the angle by . In older literature one can also find the spellings for the tangent and for the cotangent. ${\ displaystyle x}$${\ displaystyle \ tan x}$${\ displaystyle x}$${\ displaystyle \ cot x}$${\ displaystyle \ operatorname {tg} x}$${\ displaystyle \ operatorname {ctg} x}$

## definition

### Historically / geometrically

Definition on the unit circle :
${\ displaystyle {\ overline {DT}} = \ tan b \; \ {\ overline {EK}} = \ cot b}$

The term "tangent" comes from the mathematician Thomas Finck (1561–1656), who introduced it in 1583. The term “cotangent” developed from complementi tangens , i.e. the tangent of the complementary angle .

The choice of the name tangent is directly explained by the definition in the unit circle. The function values ​​correspond to the length of a tangent section:

${\ displaystyle {\ overline {DT}} = \ tan b \ qquad \ qquad {\ overline {EK}} = \ cot b}$
A right triangle, with the names of the three sides related to a variable angle α at point A and a right angle at point C.

In a right-angled triangle , the tangent of an angle is the length ratio of the opposite side to the adjacent side and the cotangent is the length ratio of the side side to the opposite side: ${\ displaystyle \ alpha}$

{\ displaystyle {\ begin {aligned} \ tan \ alpha & = {\ frac {l _ {\ text {opposite side}}} {l _ {\ text {adjacent}}}} = {\ frac {a} {b}} = {\ frac {\ sin \ alpha} {\ cos \ alpha}} \\\ cot \ alpha & = {\ frac {l _ {\ text {adjacent side}}} {l _ {\ text {opposite side}}}} = {\ frac {b} {a}} = {\ frac {\ cos \ alpha} {\ sin \ alpha}} \ end {aligned}}}

It follows immediately:

{\ displaystyle {\ begin {aligned} \ cot \ alpha & = {\ frac {1} {\ tan \ alpha}} \\\ tan \ alpha & = {\ frac {1} {\ cot \ alpha}} \ end {aligned}}}

such as

${\ displaystyle \ tan \ alpha = \ cot \ beta = \ cot (90 ^ {\ circ} - \ alpha).}$

### Formal - with a range of definitions and values

Formally, the tangent function can by means of the sine and cosine functions by

${\ displaystyle \ tan \ colon D _ {\ tan} \ to W,}$ With ${\ displaystyle \ tan x: = {\ frac {\ sin x} {\ cos x}}}$

can be defined, the range of values ​​being the real or the complex numbers depending on the application . To prevent the denominator from becoming zero, the zeros of the cosine function are omitted from the definition range : ${\ displaystyle W}$${\ displaystyle \ mathbb {R}}$${\ displaystyle \ mathbb {C}}$${\ displaystyle \ cos x}$${\ displaystyle D _ {\ tan}}$

${\ displaystyle D _ {\ tan} = \ mathbb {R} \ setminus {\ Big \ {} k \ pi + {\ frac {\ pi} {2}} \; {\ Big |} \; k \ in \ mathbb {Z} {\ Big \}}}$

in the real or

${\ displaystyle D _ {\ tan} = \ mathbb {C} \ setminus {\ Big \ {} k \ pi + {\ frac {\ pi} {2}} \; {\ Big |} \; k \ in \ mathbb {Z} {\ Big \}}}$

in the complex.

The cotangent can by analogy

${\ displaystyle \ cot \ colon D _ {\ cot} \ to W,}$ With ${\ displaystyle \ cot x: = {\ frac {\ cos x} {\ sin x}}}$

be defined, being for its domain of definition

${\ displaystyle D _ {\ cot} = \ mathbb {R} \ setminus \ {k \ pi \ mid k \ in \ mathbb {Z} \}}$

in the real or

${\ displaystyle D _ {\ cot} = \ mathbb {C} \ setminus \ {k \ pi \ mid k \ in \ mathbb {Z} \}}$

in complex results, if it is to be ensured that the denominator is not equal to zero. ${\ displaystyle \ sin x}$

For the common domain of and${\ displaystyle \ tan}$${\ displaystyle \ cot}$

${\ displaystyle \ mathbb {C} \ setminus {\ Big \ {} {\ frac {k \ pi} {2}} \; {\ Big |} \; k \ in \ mathbb {Z} {\ Big \} }}$

applies

${\ displaystyle \ cot x = {\ frac {1} {\ tan x}}.}$

## properties

Origin of the tangent function from the angular movement in the unit circle

### periodicity

The tangent and the cotangent are periodic functions with period , so it holds . ${\ displaystyle \ pi}$${\ displaystyle \ tan (x + \ pi) = \ tan (x)}$

### monotony

Tangent: Strictly increasing monotonously in the respective interval.

Cotangent: falling strictly monotonically in the respective interval.

### Symmetries

Point symmetrical to the coordinate origin:

${\ displaystyle \ tan (-x) = - \ tan x \ qquad \ qquad \ cot (-x) = - \ cot x}$

### zeropoint

 Tangent: ${\ displaystyle x = n \ cdot \ pi \,; \ quad n \ in \ mathbb {Z}}$ Cotangent: ${\ displaystyle x = \ left ({\ frac {1} {2}} + n \ right) \ cdot \ pi \,; \ quad n \ in \ mathbb {Z}}$

### Poles

 Tangent: ${\ displaystyle x = \ left ({\ frac {1} {2}} + n \ right) \ cdot \ pi \,; \ quad n \ in \ mathbb {Z}}$ Cotangent: ${\ displaystyle x = n \ cdot \ pi \,; \ quad n \ in \ mathbb {Z}}$

### Turning points

 Tangent: ${\ displaystyle x = n \ cdot \ pi \,; \ quad n \ in \ mathbb {Z}}$ Cotangent: ${\ displaystyle x = \ left ({\ frac {1} {2}} + n \ right) \ cdot \ pi \,; \ quad n \ in \ mathbb {Z}}$

Both the tangent function and the cotangent function have asymptotes, but no jumps or extremes.

## Important functional values

tangent cotangent Expression num. value
${\ displaystyle \ tan 0 ^ {\ circ}}$ ${\ displaystyle \ cot 90 ^ {\ circ}}$ ${\ displaystyle 0}$ 0
${\ displaystyle \ tan 15 ^ {\ circ}}$ ${\ displaystyle \ cot 75 ^ {\ circ}}$ ${\ displaystyle 2 - {\ sqrt {3}}}$ 0.2679491 ...
${\ displaystyle \ tan 18 ^ {\ circ}}$ ${\ displaystyle \ cot 72 ^ {\ circ}}$ ${\ displaystyle {\ sqrt {1- \ textstyle {\ frac {2} {5}} {\ sqrt {5}}}}}$ 0.3249196 ...
${\ displaystyle \ tan 22 {,} 5 ^ {\ circ}}$ ${\ displaystyle \ cot 67 {,} 5 ^ {\ circ}}$ ${\ displaystyle {\ sqrt {2}} - 1}$ 0.4142135 ...
${\ displaystyle \ tan 30 ^ {\ circ}}$ ${\ displaystyle \ cot 60 ^ {\ circ}}$ ${\ displaystyle 1 / {\ sqrt {3}}}$ 0.5773502 ...
${\ displaystyle \ tan 36 ^ {\ circ}}$ ${\ displaystyle \ cot 54 ^ {\ circ}}$ ${\ displaystyle {\ sqrt {5-2 {\ sqrt {5}}}}}$ 0.7265425 ...
${\ displaystyle \ tan 45 ^ {\ circ}}$ ${\ displaystyle \ cot 45 ^ {\ circ}}$ ${\ displaystyle 1}$ 1
${\ displaystyle \ tan 60 ^ {\ circ}}$ ${\ displaystyle \ cot 30 ^ {\ circ}}$ ${\ displaystyle {\ sqrt {3}}}$ 1.7320508 ...
${\ displaystyle \ tan 67 {,} 5 ^ {\ circ}}$ ${\ displaystyle \ cot 22 {,} 5 ^ {\ circ}}$ ${\ displaystyle {\ sqrt {2}} + 1}$ 2.4142135 ...
${\ displaystyle \ tan 75 ^ {\ circ}}$ ${\ displaystyle \ cot 15 ^ {\ circ}}$ ${\ displaystyle 2 + {\ sqrt {3}}}$ 3.7320508 ...
${\ displaystyle \ lim _ {\ alpha \ to 90 ^ {\ circ}} \ tan \ alpha}$ ${\ displaystyle \ lim _ {\ alpha \ to 0 ^ {\ circ}} \ cot \ alpha}$ ${\ displaystyle \ pm \ infty \,}$ Pole position

## Inverse function

A bijection is obtained by appropriately restricting the definition areas

tangent
${\ displaystyle \ tan \ colon \ left] - {\ frac {\ pi} {2}}, \, {\ frac {\ pi} {2}} \ right [\ to \ mathbb {R}}$.

${\ displaystyle \ operatorname {arctan} \ colon \ mathbb {R} \ to \, \ left] - {\ frac {\ pi} {2}}, \, {\ frac {\ pi} {2}} \ right [}$

is called arctangent and is therefore also bijective.

cotangent
${\ displaystyle \ cot \ colon] 0, \, \ pi [\ to \ mathbb {R}}$.

${\ displaystyle \ operatorname {arccot} \ colon \ mathbb {R} \ to \,] 0, \, \ pi [}$

is called arccotangent and is therefore also bijective.

## Series development

Tangent for | x | <½π (in radians )
tangent
The Taylor series with the development point ( Maclaurin series ) reads for${\ displaystyle x = 0}$${\ displaystyle | x | <{\ frac {\ pi} {2}}}$
{\ displaystyle {\ begin {aligned} \ tan x & = \ sum _ {n = 1} ^ {\ infty} {\ frac {(-1) ^ {n-1} \ cdot 2 ^ {2n} \ cdot \ left (2 ^ {2n} -1 \ right) \ cdot B_ {2n}} {(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 } + {\ frac {1382} {155925}} x ^ {11} + \ dotsb \ end {aligned}}}

Here, with the Bernoulli numbers called. ${\ displaystyle B_ {n}}$

cotangent
The Laurent range is for${\ displaystyle 0 <| x | <\ pi}$
{\ displaystyle {\ begin {aligned} \ cot x & = \ sum _ {n = 0} ^ {\ infty} (- 1) ^ {n} {\ frac {2 ^ {2n} B_ {2n}} {( 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} - {\ frac {2} {93555}} x ^ { 9} - \ dotsb \ end {aligned}}}

The partial fraction decomposition of the cotangene is for${\ displaystyle x \ in \ mathbb {C} \ setminus \ mathbb {Z}}$

{\ displaystyle {\ begin {aligned} \ pi \ cot \ pi x & = {\ frac {1} {x}} + \ sum _ {k = 1} ^ {\ infty} \ left ({\ frac {1} {x + k}} + {\ frac {1} {xk}} \ right) \\ & = {\ frac {1} {x}} + \ sum _ {k = 1} ^ {\ infty} {\ frac {2x} {x ^ {2} -k ^ {2}}}. \ end {aligned}}}

## Derivation

When the tangent and cotangent are derived, the otherwise rarely used trigonometric functions secant and cotangent appear :

${\ displaystyle {\ frac {\ mathrm {d}} {\ mathrm {d} x}} \ tan x = 1 + \ tan ^ {2} x = {\ frac {1} {\ cos ^ {2} x }} = \ sec ^ {2} x}$
${\ displaystyle {\ frac {\ mathrm {d}} {\ mathrm {d} x}} \ cot x = -1- \ cot ^ {2} x = - {\ frac {1} {\ sin ^ {2 } x}} = - \ csc ^ {2} x}$

The -th derivatives can be expressed with the polygamma function : ${\ displaystyle n}$

${\ displaystyle {\ frac {\ mathrm {d} ^ {n}} {\ mathrm {d} x ^ {n}}} \ tan x = {\ frac {\ psi _ {n} ({\ tfrac {1 } {2}} + {\ tfrac {x} {\ pi}}) - (- 1) ^ {n} \, \ psi _ {n} ({\ tfrac {1} {2}} - {\ tfrac {x} {\ pi}})} {\ pi ^ {n + 1}}}}$
${\ displaystyle {\ frac {\ mathrm {d} ^ {n}} {\ mathrm {d} x ^ {n}}} \ cot x = {\ frac {(-1) ^ {n} \, \ psi _ {n} (1 - {\ tfrac {x} {\ pi}}) - \ psi _ {n} ({\ tfrac {x} {\ pi}})} {\ pi ^ {n + 1}} }}$

## Antiderivatives

tangent
${\ displaystyle \ int \ tan x \, \ mathrm {d} x = - \ ln | {\ cos x} | + C}$    with   .${\ displaystyle x \ neq (2k + 1) {\ frac {\ pi} {2}}}$   ${\ displaystyle (k \ in \ mathbb {Z})}$
cotangent
${\ displaystyle \ int \ cot x \, \ mathrm {d} x = \ ln | {\ sin x} | + C}$    with   .${\ displaystyle x \ neq k \ pi}$   ${\ displaystyle (k \ in \ mathbb {Z})}$

## Complex argument

${\ displaystyle \ tan (x + \ mathrm {i} \! \ cdot \! y) = {\ frac {\ sin (2x)} {\ cos (2x) + \ cosh (2y)}} + \ mathrm {i } \; {\ frac {\ sinh (2y)} {\ cos (2x) + \ cosh (2y)}}}$   With ${\ displaystyle x, y \ in \ mathbb {R}}$
${\ displaystyle \ cot (x + \ mathrm {i} \! \ cdot \! y) = {\ frac {- \ sin (2x)} {\ cos (2x) - \ cosh (2y)}} + \ mathrm { i} \; {\ frac {\ sinh (2y)} {\ cos (2x) - \ cosh (2y)}}}$   With ${\ displaystyle x, y \ in \ mathbb {R}}$

The addition theorems for tangent and cotangent are

${\ displaystyle \ tan (x \ pm y) = {\ frac {\ tan x \ pm \ tan y} {1 \ mp \ tan x \ tan y}} \ ,, \ qquad \ cot (x \ pm y) = {\ frac {\ cot x \ cot y \ mp 1} {\ cot y \ pm \ cot x}}}$

From the addition theorems it follows in particular for double angles

${\ displaystyle \ tan (2x) = {\ frac {2 \ tan x} {1- \ tan ^ {2} x}} \ ,, \ qquad \ cot (2x) = {\ frac {\ cot ^ {2 } x-1} {2 \ cot x}}}$

## Representation of the sine and cosine using the (co) tangent

The dissolution of the identities already known from the derivation section above

${\ displaystyle {\ frac {1} {\ sin ^ {2} x}} = 1+ \ cot ^ {2} x}$
${\ displaystyle {\ frac {1} {\ cos ^ {2} x}} = 1+ \ tan ^ {2} x}$

after or when restricted to the first quadrant results in a simple one: ${\ displaystyle \ sin x}$${\ displaystyle \ cos x}$

${\ displaystyle \ sin x = {\ frac {1} {\ sqrt {1+ \ cot ^ {2} x}}}}$ For ${\ displaystyle 0
${\ displaystyle \ cos x = {\ frac {1} {\ sqrt {1+ \ tan ^ {2} x}}}}$ For ${\ displaystyle 0 \ leq x <{\ tfrac {\ pi} {2}}}$

The somewhat more complicated extensions to the whole can either be represented compactly as a limit value with the help of the floor function or more elementary by means of functions defined in sections: ${\ displaystyle \ mathbb {R}}$ ${\ displaystyle x \ mapsto \ lfloor x \ rfloor}$

${\ displaystyle \ sin x = \ lim _ {t \ to x} {\ frac {(-1) ^ {\ left \ lfloor {\ frac {t} {\ pi}} \ right \ rfloor}} {\ sqrt {1+ \ cot ^ {2} t}}} \; = {\ begin {cases} {\ frac {1} {\ sqrt {1+ \ cot ^ {2} x}}}, & {\ text { if}} \ exists k \ in \ mathbb {Z} \ colon \; 2k \ pi

${\ displaystyle \ cos x = \ lim _ {t \ to x} {\ frac {(-1) ^ {\ left \ lfloor {\ frac {t} {\ pi}} + {\ frac {1} {2 }} \ right \ rfloor}} {\ sqrt {1+ \ tan ^ {2} t}}} = {\ begin {cases} {\ frac {1} {\ sqrt {1+ \ tan ^ {2} x }}}, & {\ text {if}} \ exists k \ in \ mathbb {Z} \ colon \; (4k-1) {\ frac {\ pi} {2}}

## Rational parameterization

The tangent of the half angle can be used to describe various trigonometric functions by rational expressions: is , so is ${\ displaystyle t = \ tan {\ frac {\ alpha} {2}}}$

${\ displaystyle \ sin \ alpha = {\ frac {2t} {1 + t ^ {2}}}, \ quad \ cos \ alpha = {\ frac {1-t ^ {2}} {1 + t ^ { 2}}}, \ quad \ tan \ alpha = {\ frac {2t} {1-t ^ {2}}}.}$

In particular is

${\ displaystyle \ mathbb {R} \ to \ mathbb {R} ^ {2}, \ quad t \ mapsto \ left ({\ frac {1-t ^ {2}} {1 + t ^ {2}}} , {\ frac {2t} {1 + t ^ {2}}} \ right)}$

a parameterization of the unit circle with the exception of the point (which corresponds to the parameter ). The second point of intersection of the straight line connecting from and with the unit circle corresponds to a parameter value (see also unit circle # Rational parameterization ). ${\ displaystyle (-1,0)}$${\ displaystyle t = \ infty}$${\ displaystyle t}$${\ displaystyle (-1,0)}$${\ displaystyle (1.2t)}$

## Application: tangent and slope angle

Example of an incline

The tangent provides an important key figure for linear functions : every linear function

${\ displaystyle f \ colon \ mathbb {R} \ to \ mathbb {R}, \; x \ mapsto mx + c}$

has a straight line as a graph . The tangent of the (oriented) angle between the positive x-direction and the straight line is the slope of the straight line, i.e. H. . It doesn't matter which of the two half-straight lines you choose as the second leg. ${\ displaystyle \ alpha}$ ${\ displaystyle m}$${\ displaystyle m = \ tan \, \ alpha}$

The slope of a road is also understood to be the tangent of the slope angle. The example in the picture on the right shows a slope of 10% corresponding to a slope angle of about 5.7 ° with a tangent of 0.1.

## Application in physics

Tangent and cotangent can be used to describe the time dependence of the speed when a body is thrown upwards, if a turbulent flow is assumed for the flow resistance of the air ( Newton friction ). The coordinate system is placed in such a way that the location axis points upwards. A differential equation of the form with the gravitational acceleration g and a constant k > 0 then applies to the speed . ${\ displaystyle {\ dot {v}} = - g-kv ^ {2}}$

${\ displaystyle v (t) = v_ {g} \ cdot \ cot ({\ sqrt {gk}} t + c) \ quad {\ text {with}} \ quad c = \ operatorname {arccot} \ left ({ \ frac {v (0)} {v_ {g}}} \ right)> 0}$,

where is the limit speed that is reached in the case with air resistance . Because of the close relationships between cotangent and tangent given above, this time dependency can just as easily be expressed using the tangent: ${\ displaystyle v_ {g} = {\ sqrt {\ frac {g} {k}}}}$

${\ displaystyle v (t) = - v_ {g} \ cdot \ tan \ left ({\ sqrt {gk}} t-c '\ right) \ quad {\ text {with}} \ quad c' = \ arctan \ left ({\ frac {v (0)} {v_ {g}}} \ right)> 0}$.

This solution applies until the body has reached the highest point of its orbit (i.e. when v = 0, that is for ), after which one must use the hyperbolic tangent to describe the following case with air resistance . ${\ displaystyle t = {\ frac {\ pi / 2-c} {\ sqrt {gk}}} = {\ frac {c '} {\ sqrt {gk}}}}$

## Differential equation

The tangent is a solution to the Riccati equation

${\ displaystyle w '= 1 + w ^ {2}}$.

Factoring the right side gives

${\ displaystyle w '= 1 + w ^ {2} = (w + \ mathrm {i}) (w- \ mathrm {i})}$

with the imaginary unit . The tangent (as a complex function) has the exceptional values , : These values are not accepted because the constant functions and solutions of the differential equation and the existence and uniqueness theorem excludes that two different solutions have the same value at the same place. ${\ displaystyle \ mathrm {i}}$${\ displaystyle \ mathrm {i}}$${\ displaystyle - \ mathrm {i}}$${\ displaystyle \ mathrm {i}}$${\ displaystyle - \ mathrm {i}}$

3. For the greatest common factor this angle is ${\ displaystyle 1 {,} 5 ^ {\ circ} = {\ frac {\ pi} {120}}}$
{\ displaystyle {\ begin {aligned} \ textstyle \ tan \ left (1 {,} 5 ^ {\ circ} \ right) & = \ tan \ left ({\ frac {\ pi} {120}} \ right) = -2 + 3 {\ sqrt {2}} / 2-3 {\ sqrt {3}} / 2 - {\ sqrt {5}} + {\ sqrt {2}} {\ sqrt {3}} + { \ sqrt {2}} {\ sqrt {5}} - {\ sqrt {3}} {\ sqrt {5}} / 2 + {\ sqrt {2}} {\ sqrt {3}} {\ sqrt {5 }} / 2 \\ & \ quad + \ left (-15 / 2 + 5 {\ sqrt {2}} - 5 {\ sqrt {3}} - 7 {\ sqrt {5}} / 2 + 5 {\ sqrt {2}} {\ sqrt {3}} / 2 + 5 {\ sqrt {2}} {\ sqrt {5}} / 2-2 {\ sqrt {3}} {\ sqrt {5}} + { \ sqrt {2}} {\ sqrt {3}} {\ sqrt {5}} \ right) {\ sqrt {1-2 {\ sqrt {5}} / 5}} \\ & = 0 {,} 0261859 \ ldots \ end {aligned}}}