Trigonometry Handbook

Basic Definitions

\begin{aligned} \tan\theta & =\frac {\sin\theta}{\cos\theta} \\ \csc\theta & =\frac 1{\sin\theta} \\ \sec\theta & =\frac 1{\cos\theta} \\ \cot\theta & =\frac {\cos\theta}{\sin\theta} \end{aligned}

Parity

An even function is defined as a function that satisfies f(-x)=f(x), whereas an odd function is defined as one that satisfies f(-x)=-f(x). Even functions are symmetric about the y-axis whereas odd functions are 180^{\circ} counter-clockwise rotations about the origin.

\begin{aligned} \sin(-x) & =-\sin x \\ \cos(-x) & =\cos x \\ \tan(-x) & =-\tan x \\ \csc(-x) & =-\csc x \\ \sec(-x) & =\sec x \\ \cot(-x) & =-\cot x \end{aligned}

The only even functions are \cos x and \sec x. Similarly, for the inverse trigonometric functions, we have

\begin{aligned} \arcsin(-x) & =-\arcsin x \\ \arccos(-x) & =\arccos x \\ \arctan(-x) & =-\arctan x \\ \operatorname{arccsc}(-x) & =-\operatorname{arccsc}x \\ \operatorname{arcsec}(-x) & =\operatorname{arcsec}x \\ \operatorname{arccot}(-x) & =-\operatorname{arccot}x \end{aligned}

Unit Circle

Only the first quadrant is given, add the appropriate negative signs depending on whether the function is even or odd. See Sec. 2 to determine whether a function is even or odd.

\theta	0	\dfrac {\pi}6	\dfrac {\pi}4	\dfrac {\pi}3	\dfrac {\pi}2
\sin\theta	0	\dfrac 12	\dfrac 1{\sqrt2}	\dfrac {\sqrt3}2	1
\cos\theta	1	\dfrac {\sqrt3}2	\dfrac 1{\sqrt2}	\dfrac 12	0
\tan\theta	0	\dfrac 1{\sqrt3}	1	\sqrt3	\infty

Derivatives

Any trigonometric function that begins with the letter “c” (i.e., \cos x, \csc x, and \cot x) will have a negative sign in their derivative.

\begin{aligned} (\sin x)' & =\cos x \\ (\cos x)' & =-\sin x \\ (\tan x)' & =\sec^2 x \\ (\csc x)' & =-\csc x\cot x \\ (\sec x)' & =\sec x\tan x \\ (\cot x)' & =-\csc^2x \end{aligned}

For the inverse trigonometric functions, we also have

\begin{aligned} (\arcsin x)' & =\frac 1{\sqrt{1-x^2}} \\ (\arccos x)' & =-\frac 1{\sqrt{1-x^2}} \\ (\arctan x)' & =\frac 1{1+x^2} \\ (\operatorname{arccsc}x)' & =-\frac 1{|x|\sqrt{x^2-1}} \\ (\operatorname{arcsec}x)' & =\frac 1{|x|\sqrt{x^2-1}} \\ (\operatorname{arccot}x)' & =-\frac 1{1+x^2} \end{aligned}

The absolute value signs in |x| are necessary for \operatorname{arccsc}x and \operatorname{arcsec}x but may be ignored if only x>0 are considered.

Trigonometric Identities

Pythagorean Identities

All Pythagorean-esque identities stem from the first one in terms of \sin\theta and \cos\theta.

\begin{aligned} \sin^2\theta+\cos^2\theta & =1 \\ 1+\tan^2\theta & =\sec^2\theta \\ 1+\cot^2\theta & =\csc^2\theta \\ \sec^2\theta+\cos^2\theta & =\sec^2\theta\csc^2\theta \end{aligned}

Angle Sum Formulas

For the standard trigonometric functions, these angles relate the angle sum or difference, \theta\pm\varphi, with the individual components, \theta and \varphi.

\begin{aligned} \sin(\theta\pm\varphi) & =\sin\theta\cos\varphi\pm\cos\theta\sin\varphi \\ \cos(\theta\pm\varphi) & =\cos\theta\cos\varphi\mp\sin\theta\sin\varphi \\ \tan(\theta\pm\varphi) & =\frac {\tan\theta\pm\tan\varphi}{1\mp\tan\theta\tan\varphi} \\ \csc(\theta\pm\varphi) & =\frac {\sec\theta\sec\varphi\csc\theta\csc\varphi}{\sec\theta\csc\varphi\pm\csc\theta\sec\varphi} \\ \sec(\theta\pm\varphi) & =\frac {\sec\theta\sec\varphi\csc\theta\csc\varphi}{\csc\theta\csc\varphi\mp\sec\theta\sec\varphi} \\ \cot(\theta\pm\varphi) & =\frac {\cot\theta\cot\varphi\mp1}{\cot\varphi\pm\cot\theta} \end{aligned}

For the inverse trigonometric functions, we also have

\begin{aligned} \arcsin\theta\pm\arcsin\varphi & =\arcsin\left(\theta\sqrt{1-\varphi^2}\pm\varphi\sqrt{1-\theta^2}\right) \\ \arccos\theta\pm\arccos\varphi & =\arccos\left(\theta\varphi\mp\sqrt{\left(1-\theta^2\right)\left(1-\varphi^2\right)}\right) \\ \arctan\theta\pm\arctan\varphi & =\arctan\left(\frac {\theta\pm\varphi}{1\mp\theta\varphi}\right) \\ \mathrm{arccot}\theta\pm\operatorname{arccot}\varphi & =\operatorname{arccot}\left(\frac {\theta\varphi\pm1}{\varphi\pm\theta}\right) \end{aligned}

Double Angle Identities

These formulas can also be used to reduce

\begin{aligned} \sin2\theta & =2\sin\theta\cos\theta & =\frac {2\tan\theta}{1+\tan^2\theta} \\ & =(\sin\theta+\cos\theta)^2-1 \\ \cos2\theta & =\cos^2\theta-\sin^2\theta & =\frac {1-\tan^2\theta}{1+\tan^2\theta} \\ & =2\cos^2\theta-1 \\ & =1-2\sin^2\theta \\ \tan2\theta & =\frac {2\tan\theta}{1-\tan^2\theta} \\ \csc2\theta & =\frac 12\sec\theta\csc\theta & =\frac {1+\tan^2\theta}{2\tan\theta} \\ \sec2\theta & =\frac {\sec^2\theta}{2-\sec^2\theta} & =\frac {1+\tan^2\theta}{1-\tan^2\theta} \\ \cot2\theta & =\frac {\cot^2\theta-1}{2\cot\theta} & =\frac {1-\tan^2\theta}{2\tan\theta} \end{aligned}

Triple Angle Identities

Taylor Series

Only the Taylor series with elementary coefficients are listed here. Series expansions exist for other functions like \tan x and \sec x, but their coefficients are only expressible in terms of “fancy” sequences like Bernoulli numbers and Euler numbers.

\begin{aligned} \sin x & =\sum\limits_{k=0}^{+\infty}\frac {(-1)^k}{(2k+1)!}x^{2k+1} \\ \cos x & =\sum\limits_{k=0}^{+\infty}\frac {(-1)^k}{(2k)!}x^{2k} \\ \arcsin x & =\sum\limits_{k=0}^{+\infty}\frac {(2k)!}{4^kk!^2(2k+1)}x^{2k+1} \\ \arccos x & =\frac {\pi}2-\sum\limits_{k=0}^{+\infty}\frac {(2k)!}{4^kk!^2(2k+1)}x^{2k+1} \\ \arctan x & =\sum\limits_{k=0}^{+\infty}\frac {(-1)^k}{2k+1}x^{2k+1} \end{aligned}