A trigonometrical identity is a relation implying of the goniometrical functions and which is checked for all the values of the variables intervening in the relation. These identities can be useful when an expression comprising of the goniometrical functions needs to be simplified. They thus constitute a useful “toolbox” for the solution to problem.

The goniometrical functions are useful much in Intégration, to integrate “nontrigonometrical” functions: a usual process consists in carrying out a change of variable by using a goniometrical function, and then simplifying the integral obtained with the trigonometrical identities.

Notation : with the goniometrical functions, we will define sin², cos², etc, the functions such as for any reality X , sin² ( X ) = (sin ( X ))²,…

Starting from the definitions

$\backslash \; tan\; (X)\; =\; \backslash \; frac\; \{\backslash \; sin\; (X)\}\; \{\backslash \; cos\; (X)\}\; \backslash \; qquad\; \backslash \; operatorname\; \{cotan\}\; (X)\; =\; \backslash \; frac\; \{1\}\; \{\backslash \; tan\; (X)\}\; =\; \backslash \; frac\; \{\backslash \; cos\; (X)\}\; \{\backslash \; sin\; (X)\}$

$\backslash \; operatorname\; \{dry\}\; (X)\; =\; \backslash \; frac\; \{1\}\; \{\backslash \; cos\; (X)\}\; \backslash \; qquad\; \backslash \; operatorname\; \{cosec\}\; (X)\; =\; \backslash \; frac\; \{1\}\; \{\backslash \; sin\; (X)\}$

Properties related to the trigonometrical circle

Periodicity:

$\backslash \; sin\; (X)\; =\; \backslash \; sin\; (X\; +\; 2\; \backslash \; pi)\; \backslash \; qquad\; \backslash \; cos\; (X)\; =\; \backslash \; cos\; (X\; +\; 2\; \backslash \; pi)\; \backslash \; qquad\; \backslash \; tan\; (X)\; =\; \backslash \; tan\; (X\; +\; \backslash \; pi)$

Parity, disparity:

$\backslash \; sin\; (-\; X)\; =\; -\; \backslash \; sin\; (X)\; \backslash \; qquad\; \backslash \; cos\; (-\; X)\; =\; \backslash \; cos\; (X)$

$\backslash \; tan\; (-\; X)\; =\; -\; \backslash \; tan\; (X)\; \backslash \; qquad\; \backslash \; operatorname\; \{cotan\}\; (-\; X)\; =\; -\; \backslash \; operatorname\; \{cotan\}\; (X)$

Symmetries:

$\backslash \; sin\; (X)\; =\; \backslash \; cos\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; pi\}\; \{2\}\; -\; X\; \backslash \; right)$

\ qquad \ cos (X) = \ sin \ left (\ frac {\ pi} {2} - X \ right) \ qquad \ tan (X) = \ operatorname {cotan} \ left (\ frac {\ pi} {2} - X \ right)

$\backslash \; sin\; (X\; +\; \backslash \; pi)\; =\; -\; \backslash \; sin\; (X)\; \backslash \; qquad\; \backslash \; cos\; (X\; +\; \backslash \; pi)\; =\; -\; \backslash \; cos\; (X)\; \backslash \; qquad\; \backslash \; tan\; (X\; +\; \backslash \; pi)\; =\; \backslash \; tan\; (X)$

$\backslash \; sin\; (\backslash \; pi\; -\; X)\; =\; \backslash \; sin\; (X)\; \backslash \; qquad\; \backslash \; cos\; (\backslash \; pi\; -\; X)\; =\; -\; \backslash \; cos\; (X)\; \backslash \; qquad\; \backslash \; tan\; (\backslash \; pi\; -\; X)\; =\; -\; \backslash \; tan\; (X)$

Rotations

$\backslash \; sin\; \backslash \; left\; (X\; +\; \backslash \; frac\; \{\backslash \; pi\}\; \{2\}\; \backslash \; right)\; =\; \backslash \; cos\; (X)$

\ qquad \ cos \ left (X + \ frac {\ pi} {2} \ right) = - \ sin (X) \ qquad \ tan \ left (X + \ frac {\ pi} {2} \ right) = - \ operatorname {cotan} (X)

$\backslash \; sin\; \backslash \; left\; (X\; -\; \backslash \; frac\; \{\backslash \; pi\}\; \{2\}\; \backslash \; right)\; =\; -\; \backslash \; cos\; (X)$

\ qquad \ cos \ left (X - \ frac {\ pi} {2} \ right) = \ sin (X) \ qquad \ tan \ left (X - \ frac {\ pi} {2} \ right) = - \ operatorname {cotan} (X)

Starting from the Theorem of Pythagore

$\backslash \; sin^2\; (X)\; +\; \backslash \; cos^2\; (X)\; =\; 1\; \backslash \; qquad\; \backslash \; tan^2\; (X)\; +\; 1\; =\; \backslash \; sec^2\; (X)\; \backslash \; qquad\; \{\backslash \; rm\; cotan\}\; ^2\; (X)\; +\; 1\; =\; \{\backslash \; rm\; cosec\}\; ^2\; (X)$

Trigonometrical equation

$\backslash \; cos\; X\; =\; \backslash \; cos\; has\; \backslash \; Leftrightarrow\; x=a+2k\; \backslash \; pi\; \backslash \; quad\; or\; \backslash \; quad\; x=-a+2k\; \backslash \; pi\; \backslash \; qquad\; (K\; \backslash \; in\; \backslash \; mathbb\; \{Z\})$

$\backslash \; sin\; X\; =\; \backslash \; sin\; has\; \backslash \; Leftrightarrow\; x=a+2k\; \backslash \; pi\; \backslash \; quad\; or\; \backslash \; quad\; x=\; \backslash \; pi-a+2k\; \backslash \; pi\; \backslash \; qquad\; (K\; \backslash \; in\; \backslash \; mathbb\; \{Z\})$

$\backslash \; tan\; X\; =\; \backslash \; tan\; has\; \backslash \; Leftrightarrow\; x=a+k\; \backslash \; pi\; \backslash \; qquad\; (K\; \backslash \; in\; \backslash \; mathbb\; \{Z\})$

Formulas of addition and difference

The fastest means to find these formulas is to use the formulas of Euler in analysis complexes.

$\backslash \; sin\; (has\; +\; b)\; =\; \backslash \; sin\; has\; \backslash \; cos\; B\; +\; \backslash \; cos\; has\; \backslash \; sin\; B\; \backslash ,$

$\backslash \; sin\; (has\; -\; b)\; =\; \backslash \; sin\; has\; \backslash \; cos\; B\; -\; \backslash \; cos\; has\; \backslash \; sin\; B\; \backslash ,$

$\backslash \; cos\; (has\; +\; b)\; =\; \backslash \; cos\; has\; \backslash \; cos\; B\; -\; \backslash \; sin\; has\; \backslash \; sin\; B\; \backslash ,$

$\backslash \; cos\; (has\; -\; b)\; =\; \backslash \; cos\; has\; \backslash \; cos\; B\; +\; \backslash \; sin\; has\; \backslash \; sin\; B\; \backslash ,$

A Mnemotechnical means to retain: “The cosine is malicious: he does not sympathize with the sines, and moreover he changes the signs”.

$\backslash \; tan\; (has\; +\; b)\; =\; \backslash \; frac\; \{\backslash \; tan\; has\; +\; \backslash \; tan\; B\}\; \{1\; -\; \backslash \; tan\; has\; \backslash \; tan\; B\}\; \backslash ,$

$\backslash \; tan\; (has\; -\; b)\; =\; \backslash \; frac\; \{\backslash \; tan\; has\; -\; \backslash \; tan\; B\}\; \{1\; +\; \backslash \; tan\; has\; \backslash \; tan\; B\}\; \backslash ,$

An interesting consequence of these equalities is that they make it possible to bring back the linear combination of a sine and a cosine to a sine:

$\backslash \; alpha\; \backslash \; sin\; wx+\; \backslash \; beta\; \backslash \; cos\; wx=\; \backslash \; sqrt\; \{\backslash \; alpha^2+\; \backslash \; beta^2\}\; \backslash \; cdot\; \backslash \; sin\; (wx+\; \backslash \; varphi)$

where

$\backslash \; varphi=\; \{\backslash \; rm\; arctan\}\; (\backslash \; beta\; \backslash \; alpha)$ if α is positive and $\backslash \; varphi=\; \{\backslash \; rm\; arctan\}\; (\backslash \; beta\; \backslash \; alpha)\; +\; \backslash \; pi$ if not

Formulas of duplication and angle half

Formulas of the double angle

Also called formulas of angle doubles , it can be obtained by replacing has and B by X in the formulas of addition and by using the Théorème of Pythagore for the two last, or by using the Formule of Moivre with $n\; =\; 2\; \backslash ,\; \backslash !$.

$\backslash \; sin\; 2x\; =\; 2\; \backslash \; sin\; X\; \backslash \; cos\; X\; \backslash ,$

$\backslash \; cos\; 2x=\; \backslash \; cos^2\; X\; -\; \backslash \; sin^2\; X\; =\; 2\; \backslash \; cos^2\; x-1\; =\; 1-2\; \backslash \; sin^2\; X\; \backslash ,$

$\backslash \; tan\; 2x\; =\; \{2\; \backslash \; tan\; X\; \backslash \; over\; 1\; -\; \backslash \; tan^2\; X\}\; =\; \{2\; \backslash \; cot\; X\; \backslash \; over\; \backslash \; cot^2\; X\; 1\}\; =\; \{2\; \backslash \; over\; \backslash \; cot\; X\; -\; \backslash \; tan\; X\}\; \backslash ,$

Formulas of reduction of the square

These formulas make it possible to write $\backslash \; cos^2\; (X)$, $\backslash \; sin^2\; (X)$ and $\backslash \; tan^2\; (X)$ according to the cosine of the double angle.

$\backslash \; cos^2\; (X)\; =\; \{1\; +\; \backslash \; cos\; (2x)\; \backslash \; over\; 2\}$

$\backslash \; sin^2\; (X)\; =\; \{1\; -\; \backslash \; cos\; (2x)\; \backslash \; over\; 2\}$

$\backslash \; tan^2\; (X)\; =\; \{1\; -\; \backslash \; cos\; (2x)\; \backslash \; over\; 1\; +\; \backslash \; cos\; (2x)\}$

Formulas of angle half

By replacing $x~$ by $\backslash \; frac\; \{X\}\; \{2\}$ in the formulas of reduction of the squares, and then by seeking the expression of $\backslash \; cos\; \backslash \; left\; (\backslash \; frac\; x2\; \backslash \; right)$ and $\backslash \; sin\; \backslash \; left\; (\backslash \; frac\; x2\; \backslash \; right)$, we obtain:

$\backslash \; left|\backslash \; cos\; \backslash \; left\; (\backslash \; frac\; \{X\}\; \{2\}\; \backslash \; right)\; \backslash \; right|\; =\; \backslash \; sqrt\; \{\backslash \; left\; (\backslash \; frac\; \{1\; +\; \backslash \; cos\; (X)\}\{2\}\; \backslash \; right)\}$

$\backslash \; left|\backslash \; sin\; \backslash \; left\; (\backslash \; frac\; \{X\}\; \{2\}\; \backslash \; right)\; \backslash \; right|\; =\; \backslash \; sqrt\; \{\backslash \; left\; (\backslash \; frac\; \{1\; -\; \backslash \; cos\; (X)\}\{2\}\; \backslash \; right)\}$

By multiplying $\backslash \; tan\; \backslash \; left\; (\backslash \; frac\; x2\; \backslash \; right)$ by $\backslash \; frac\; \{2\; \backslash \; cos\; (x/2)\}\{2\; \backslash \; cos\; (x/2)\}$ and by replacing it by $\backslash \; frac\; \{\backslash \; sin\; (x/2)\}\{\backslash \; cos\; (x/2)\}$ one obtains with the numerator $\backslash \; sin\; (X)\; ~$ according to the double formula of angle, and with the denominator $2\; \backslash \; cos^2\; \backslash \; left\; (\backslash \; frac\; x2\; \backslash \; right)$ which is also equal to $\backslash \; cos\; (X)\; +1~$ according to the formula of reduction of the square.

The second formula comes from the first by multiplying numerator and denominator by $\backslash \; sin\; (X)\; ~$ and while simplifying by using the theorem of Pythagore.

$\backslash \; tan\; \backslash \; left\; (\backslash \; frac\; \{X\}\; \{2\}\; \backslash \; right)\; =\; \backslash \; frac\; \{\backslash \; sin\; (X)\}\{\backslash \; cos\; (X)\; +\; 1\}\; =\; \backslash \; frac\; \{1\; -\; \backslash \; cos\; (X)\}\{\backslash \; sin\; (X)\}$

Formulas implying the “tangent of the arc half”

If one poses $t=\; \backslash \; tan\; \backslash \; left\; (\backslash \; frac\; \{X\}\; \{2\}\; \backslash \; right)$, one a:

$\backslash \; cos\; (X)\; =\; \backslash \; frac\; \{1-t^2\}\; \{1+t^2\}$

$\backslash \; sin\; (X)\; =\; \backslash \; frac\; \{2t\}\; \{1+t^2\}$

$\backslash \; tan\; (X)\; =\; \backslash \; frac\; \{2t\}\; \{1-t^2\}$

In the case of change of variable in integration, one will add:

$D\; X\; =\; \backslash \; frac\; \{2d\; T\}\; \{1\; +\; t^2\}$

These formulas make it possible to simplify trigonometrical calculations while being brought back to calculations on rational fractions. They also make it possible to determine the whole of the rational points of the circle unit.

Formulas of Simpson

Transformation of products into sums

$\backslash \; cos\; p\; \backslash \; cos\; q=\; \backslash \; frac\; \{1\}\; \{2\}$

$\backslash \; sin\; p\; \backslash \; cos\; q=\; \backslash \; frac\; \{1\}\; \{2\}$

$\backslash \; sin\; p\; \backslash \; sin\; q=\; \backslash \; frac\; \{1\}\; \{2\}$

These formulas can be shown by developing their members of right-hand side by using the formulas of addition

Transformation of sums into products

$\backslash \; sin\; p\; +\; \backslash \; sin\; Q\; =\; 2\; \backslash \; sin\; \backslash \; frac\; \{p+q\}\; \{2\}\; \backslash \; cos\; \backslash \; frac\; \{p-q\}\; \{2\}$

$\backslash \; sin\; p\; -\; \backslash \; sin\; Q\; =\; 2\; \backslash \; cos\; \backslash \; frac\; \{p+q\}\; \{2\}\; \backslash \; sin\; \backslash \; frac\; \{p-q\}\; \{2\}$

$\backslash \; cos\; p\; +\; \backslash \; cos\; Q\; =\; 2\; \backslash \; cos\; \backslash \; frac\; \{p+q\}\; \{2\}\; \backslash \; cos\; \backslash \; frac\; \{p-q\}\; \{2\}$

$\backslash \; cos\; p\; -\; \backslash \; cos\; Q\; =\; -2\; \backslash \; sin\; \backslash \; frac\; \{p+q\}\; \{2\}\; \backslash sin\; \backslash \; frac\; \{p-q\}\; \{2\}$

It is enough to replace p by $\backslash \; frac\; \{p+q\}\; \{2\}$ and Q by $\backslash \; frac\; \{p-q\}\; \{2\}$ in the formulas of transformation of product all in all.

Average a mnemotechnics to retain: “If, coconut, if; coconut, if! Priority with the sine and the addition, -2 with the last”.

$\backslash \; tan\; (p)\; +\; \backslash \; tan\; (Q)\; =\; \backslash \; frac\; \{\backslash \; sin\; (p+q)\}\{\backslash \; cos\; (p)\; \backslash ,\; \backslash \; cos\; (Q)\}$

Formulas of Euler

$\backslash \; cos\; x=\; \backslash \; frac\; \{e^\; \{ix\}\; +e^\; \{-\; ix\}\}\; \{2\}$

$\backslash \; sin\; x=\; \backslash \; frac\; \{e^\; \{ix\}\; -\; e^\; \{-\; ix\}\}\; \{2i\}$

detailed Article: Formulas of Euler See also: Trigonométrie complexes

Formulate of Moivre and formulas of multiple angle

The Formule of Moivre is written:

$\backslash \; cos\; (nx)\; +i\; \backslash \; sin\; (nx)\; =\; (\backslash \; cos\; (X)\; +i\; \backslash \; sin\; (X))^n\; \backslash ,\; \backslash !$ where I is the imaginary number.

If T_n is N ième polynomial of Tchebychev then

$\backslash \; cos\; (nx)\; =T\_n\; (\backslash \; cos\; (X))\backslash ,\; \backslash !.$

The Noyau of Dirichlet D_n is the function defined by:

for any reality X , $D\_n\; (X)\; =1+2\; \backslash \; cos\; (X)\; +2\; \backslash \; cos\; (2x)\; +2\; \backslash \; cos\; (3x)\; +\; \backslash \; cdots+2\; \backslash \; cos\; (nx)\; =\; \backslash \; frac\; \{\backslash \; sin\; \backslash \; left\; (\backslash \; left\; (n+\; \backslash \; frac\; \{1\}\; \{2\}\; \backslash \; right)\; X\; \backslash \; right)\}\{\backslash \; sin\; (x/2)\}$

The Produces convolution of any function of integrable square and of period 2π with the core of Dirichlet coincides with the sum of order N of its Fourier series.

Linearization

$\backslash \; cos^\; \{N\}\; X\; =\; \backslash \; left\; (\backslash \; frac\; \{e^\; \{ix\}\; +e^\; \{-\; ix\}\}\; \{2\}\; \backslash \; right)\; ^\; \{N\}$

$\backslash \; sin^\; \{N\}\; x=\; \backslash \; left\; (\backslash \; frac\; \{e^\; \{ix\}\; -\; e^\; \{-\; ix\}\}\; \{2i\}\; \backslash \; right)\; ^\; \{N\}$

And like the theorem De Moivre:

$(\backslash \; cos\; (\backslash \; theta)\; +\; I\; \backslash \; sin\; (\backslash \; theta))^n\; =\; \backslash \; cos\; (N\; \backslash \; theta)\; +\; I\; \backslash \; sin\; (N\; \backslash \; theta)\; \backslash ,$

It is then enough to develop the sum thanks to the Formule of the binomial theorem, to gather the terms knowing that

$e^\; \{I\; (n-k)\; X\}\; e^\; \{-\; ikx\}\; =\; e^\; \{I\; (n-2k)\; X\}\; \backslash ,$

$e^\; \{imx\}\; +\; e^\; \{-\; imx\}\; =\; 2\; \backslash \; cos\; (MX)\; \backslash ,$

$e^\; \{imx\}\; -\; e^\; \{-\; imx\}\; =\; 2i\; \backslash \; sin\; (MX)\; \backslash ,$

Formules of linearization of degree 2

$\backslash \; cos^2\; has\; +\; \backslash \; sin^2\; has\; =\; 1$

$\backslash \; cos^2\; has\; =$

$\backslash \; sin^2\; has\; =$

$\backslash \; tan^2\; has\; =$

Reciprocal goniometrical functions

They are the reciprocal functions of the functions sin, cos and tan.

$y=\; \backslash \; arcsin\; X\; \backslash \; Leftrightarrow\; x=\; \backslash \; sin\; there\; \backslash \; mathrm\; \{\backslash \; with\; \backslash \}\; there\; \backslash \; in\; \backslash \; left$

$y=\; \backslash \; arccos\; X\; \backslash \; Leftrightarrow\; x=\; \backslash \; cos\; there\; \backslash \; mathrm\; \{\backslash \; with\; \backslash \}\; there\; \backslash \; in\; \backslash \; left$

$y=\; \backslash \; arctan\; X\; \backslash \; Leftrightarrow\; x=\; \backslash \; tan\; there\; \backslash \; mathrm\; \{\backslash \; with\; \backslash \}\; there\; \backslash \; in\; \backslash \; left]\; \backslash \; frac\; \{-\; \backslash \; pi\}\; \{2\}\; \backslash ;\; \backslash \; frac\; \{\backslash \; pi\}\; \{2\}\; \backslash \; right$

If X > 0 then

$\backslash \; operatorname\; \{Arctan\}\; (X)\; +\; \backslash \; operatorname\; \{Arctan\}\; (1/x)\; =\; \backslash \; frac\; \{\backslash \; pi\}\; \{2\}.$

If X < the 0 then right-sided of the equality is equal to $-\; \backslash \; frac\; \{\backslash \; pi\}\; \{2\}$.

$\backslash \; operatorname\; \{Arctan\}\; (X)\; +\; \backslash \; operatorname\; \{Arctan\}\; (there)\; =\; \backslash \; operatorname\; \{Arctan\}\; \backslash \; left\; (\backslash \; frac\; \{x+y\}\; \{1-xy\}\; \backslash \; right)$

Many identities similar to following can be obtained starting from the Théorème of Pythagore:

$\backslash \; cos\; (\backslash \; operatorname\; \{Arcsin\}\; (X))=\; \backslash \; sqrt\; \{1-x^2\}$

Formulate sines

$\backslash \; frac\; \{has\}\; \{sinA\}\; =\; \backslash \; frac\; \{B\}\; \{sinB\}\; =\; \backslash \; frac\; \{C\}\; \{sinC\}\; =\; \backslash \; frac\; \{ABC\}\; \{2S\}\; =\; \backslash \; frac\; \{ABC\}\; \{2pr\}\; =\; 2R$

It is enough to see that ha = 2S with H = C .sinB thus ABC. sin B = 2S.b. In addition the point of contest of the bisectrices R is I the circle inscribed ruffle +rb + rc = 2S = r.2p.

R is the radius of the circumscribed circle.

Formulate differences on the sides

• $b\; -\; C\; =\; has\; \backslash \; cdot$

It is a direct application of the formula of the sines and formulas of factorization. One in the same way has also the sum on the sides. For an angle has sufficiently small, B is not very different from C, and the difference is worth:

$b\; -\; C\; =\; -\; has\; \backslash \; cos\; B$,

This result is obtained by using the Al-Kashi formula for the angle B then by neglecting $a^2$ or by considering the projection of BC on AB.

• Formula of the tangents (known as sometimes formula of the land-surveyors):

$=\; \backslash \; cfrac\; \{\backslash \; tan\; \backslash \; left\; (\backslash \; frac\; \{B-C\}\; \{2\}\; \backslash \; right)\}\{\backslash \; tan\; \backslash \; left\; (\backslash \; frac\; \{B+C\}\; \{2\}\; \backslash \; right)\}$

$\backslash \; tan\; \backslash \; left\; (\backslash \; cfrac\; \{has\}\; \{2\}\; \backslash \; right)\; =\; \{R\; \backslash \; over\; \{Pa\}\}$

(to remember that $\backslash \; cfrac\; \{has\}\; \{2\}$ being acute, $\backslash \; tan\; \backslash \; left\; (\backslash \; cfrac\; \{has\}\; \{2\}\; \backslash \; right)\; =\; \backslash \; sqrt\; \{\backslash \; cfrac\; \{1\; \backslash \; cos\; (A)\}\; \{1+\; \backslash \; cos\; (A)\}\}$ and to apply Al-Kashi)

Identities without variable

Richard Feynman which was famous to have learned very well its formulas from trigonometry, always remembered of this curious identity:

$\backslash cos\; 20^\backslash circ\backslash cdot\backslash cos\; 40^\backslash circ\backslash cdot\backslash cos\; 80^\backslash circ=1/8.$

Such an identity is an example of identity which does not contain a variable and is obtained starting from the equality:

$\backslash \; prod\_\; \{j=0\}\; ^\; \{k-1\}\; \backslash \; cos\; (2^j\; X)\; =\; \backslash \; frac\; \{\backslash \; sin\; (2^k\; X)\}\{2^k\; \backslash \; sin\; (X)\}.$

The following relations can also be regarded as identities without variable:

$\backslash cos\; 36^\backslash circ+\backslash cos\; 108^\backslash circ=1/2.$

$\backslash cos\; 24^\backslash circ+\backslash cos\; 48^\backslash circ+\backslash cos\; 96^\backslash circ+\backslash cos\; 168^\backslash circ=1/2.$

It is that the measurement in degrees of the angles does not give a formula simpler than with measurement in radians when we consider this identity with 21 with the denominators:

$\backslash \; cos\; \backslash \; frac\; \{2\; \backslash \; pi\}\; \{21\}\; +\; \backslash \; cos\; \backslash \; frac\; \{2\; (2\; \backslash \; pi)\}\{21\}\; +\; \backslash \; cos\; \backslash \; frac\; \{4\; (2\; \backslash \; pi)\}\{21\}\; +\; \backslash \; cos\; \backslash \; frac\; \{5\; (2\; \backslash \; pi)\}\{21\}\; +\; \backslash \; cos\; \backslash \; frac\; \{8\; (2\; \backslash \; pi)\}\{21\}\; +\; \backslash \; cos\; \backslash \; frac\; \{10\; (2\; \backslash \; pi)\}\{21\}\; =1/2.$

But factors 1,2,4,5,8,10 can make us think of the entireties lower than 21/2 which do not have a common factor with 21. The last examples are consequences of a basic result on the cyclotomic polynomials; the cosine are the real parts of the roots of these polynomials; the sum of the zeros gives the value of the Fonction of Möbius into 21 (in the very last case which precedes); only the half of the roots are present in the preceding relation.

In analyzes…

In analysis, it is essential that the angles which appear as goniometrical arguments of functions are measured in Radian S; if they are measured in degrees or in any other unit, then the relations deferred below become false. If the goniometrical functions are geometrically defined, then their derivative can be obtained by establishing these limits beforehand:

$\backslash \; lim\_\; \{X\; \backslash \; rightarrow\; 0\}\; \backslash \; frac\; \{\backslash \; sin\; (X)\}\{X\}\; =1,$

and

$\backslash \; lim\_\; \{X\; \backslash \; rightarrow\; 0\}\; \backslash \; frac\; \{1\; \backslash \; cos\; (X)\}\{x^2\}\; =\; \backslash \; frac\; \{1\}\; \{2\},$

and by then using the definition with the limits of derived in a point as well as the theorems from addition; if the goniometrical functions are defined by their Taylor series, then the derivative can be obtained by deriving the whole series term in the long term.

$\{D\; \backslash \; sin\; \backslash \; over\; dx\}\; (X)\; =\; \backslash \; cos\; (X)\; =\; \backslash \; sin\; \backslash \; left\; (X\; +\; \backslash \; frac\; \backslash \; pi\; \{2\}\; \backslash \; right)$

The other goniometrical functions can be derived by using the preceding identities and the rules from Dérivation, for example:

$\{D\; \backslash \; cos\; \backslash \; over\; dx\}\; (X)\; =\; -\; \backslash \; sin\; (X)\; =\; \backslash \; cos\; \backslash \; left\; (X\; +\; \backslash \; frac\; \backslash \; pi\; \{2\}\; \backslash \; right)$

$\{D\; \backslash \; tan\; \backslash \; over\; dx\}\; (X)\; =\; 1\; +\; \backslash \; tan^2\; (X)\; =\; \backslash \; frac\; \{1\}\; \{\backslash \; cos^2\; (X)\}\; =\; \backslash \; sec^2\; (X)$

$\{D\; \backslash \; operatorname\; \{Arcsin\}\; \backslash \; over\; dx\}\; (X)\; =\; \backslash \; frac\; \{1\}\; \{\backslash \; sqrt\; \{1-x^2\}\}$

$\{D\; \backslash \; operatorname\; \{Arctan\}\; \backslash \; over\; dx\}\; (X)\; =\; \backslash \; frac\; \{1\}\; \{1+x^2\}$

The identities on the integrals can be found in the table of integrals.

See too

• goniometrical Trigonometry
• Function
• Formulas of Euler
• Formula of hyperbolic Moivre
• Function

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