Trigonometry
Outline History Usage Functions (sin, cos, tan, inverse) Generalized trigonometry
Reference
Identities Exact constants Tables Unit circle
Laws and theorems
Sines Cosines Tangents Cotangents Pythagorean theorem
Calculus
Trigonometric substitution Integrals (inverse functions) Derivatives Trigonometric series
Mathematicians
Hipparchus Ptolemy Brahmagupta al-Hasib al-Battani Regiomontanus Viète de Moivre Euler Fourier
v t e

Function	Derivative
${\displaystyle \sin(x)}$	${\displaystyle \cos(x)}$
${\displaystyle \cos(x)}$	${\displaystyle -\sin(x)}$
${\displaystyle \tan(x)}$	${\displaystyle \sec ^{2}(x)}$
${\displaystyle \cot(x)}$	${\displaystyle -\csc ^{2}(x)}$
${\displaystyle \sec(x)}$	${\displaystyle \sec(x)\tan(x)}$
${\displaystyle \csc(x)}$	${\displaystyle -\csc(x)\cot(x)}$
${\displaystyle \arcsin(x)}$	${\displaystyle {\frac {1}{\sqrt {1-x^{2}}}}}$
${\displaystyle \arccos(x)}$	${\displaystyle -{\frac {1}{\sqrt {1-x^{2}}}}}$
${\displaystyle \arctan(x)}$	${\displaystyle {\frac {1}{x^{2}+1}}}$
${\displaystyle \operatorname {arccot}(x)}$	${\displaystyle -{\frac {1}{x^{2}+1}}}$
${\displaystyle \operatorname {arcsec}(x)}$	${\displaystyle {\frac {1}{|x|{\sqrt {x^{2}-1}}}}}$
${\displaystyle \operatorname {arccsc}(x)}$	${\displaystyle -{\frac {1}{|x|{\sqrt {x^{2}-1}}}}}$

The differentiation of trigonometric functions is the mathematical process of finding the derivative of a trigonometric function, or its rate of change with respect to a variable. For example, the derivative of the sine function is written sin′(a) = cos(a), meaning that the rate of change of sin(x) at a particular angle x = a is given by the cosine of that angle.

All derivatives of circular trigonometric functions can be found from those of sin(x) and cos(x) by means of the quotient rule applied to functions such as tan(x) = sin(x)/cos(x). Knowing these derivatives, the derivatives of the inverse trigonometric functions are found using implicit differentiation.

Proofs of derivatives of trigonometric functions

Limit of sin(θ)/θ as θ tends to 0

The diagram at right shows a circle with centre O and radius r = 1. Let two radii OA and OB make an arc of θ radians. Since we are considering the limit as θ tends to zero, we may assume θ is a small positive number, say 0 < θ < ½ π in the first quadrant.

In the diagram, let R₁ be the triangle OAB, R₂ the circular sector OAB, and R₃ the triangle OAC.

The area of triangle OAB is:

${\displaystyle \mathrm{A}\mathrm{r}\mathrm{e}\mathrm{a}(R_1)=\frac{1}{2}|OA||OB|\sin <!--\; \; -->\mathrm{\theta <!--\; \theta \; -->}=\frac{1}{}}$Zdroj:https://en.wikipedia.org?pojem=Differentiation_of_trigonometric_functions
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