4.1.1 Unit Circle
The trig functions are the functions of rotation on a circumference. The triangular
right angle definitions are a specific case.

$$\sin(\theta)=y/r\qquad\csc(\theta)=r/y$$
$$\cos(\theta)=x/r\qquad\sec(\theta)=r/x$$
$$\tan(\theta)=y/x\qquad\cot(\theta)=x/y$$
$$\theta=\sin^{-1}(y/r)\qquad\theta=\csc^{-1}(r/y)$$
$$\theta=\cos^{-1}(x/r)\qquad\theta=\sec^{-1}(r/x)$$
$$\theta=\tan^{-1}(y/x)\qquad\theta=\cot^{-1}(x/y)$$
4.1.2 Inverse, Reciprocal & Power Notation
Sources reference inverses like $\arcsin(θ)$ to avoid confusion. Here they are listed as above to save space, while reciprocal functions are as
$$\sin(\theta)^{-1}=\frac{1}{\sin(\theta)}=\csc(\theta)≠\sin^{-1}(\theta)$$
However, powers other than $-1$ always mean the same in either of the following
$$\sin(\theta)^n=\sin^n(\theta),\medspace\forall n≠-1$$
4.1.3 Even/Odd Identities & Reflections
$$\sin(-\theta)=-\sin(\theta)\qquad\cos(-\theta)=+\cos(\theta)$$
$$\csc(-\theta)=-\csc(\theta)\qquad\sec(-\theta)=+\sec(\theta)$$
$$\tan(-\theta)=-\tan(\theta)\qquad\cot(-\theta)=-\cot(\theta)$$
The yielded sign value depends on which quadrant are the values of $(x,y)$
$\text{II: $\pi/2\lt 0\le\pi$}$
|
$\text{I: $0\le\theta\le\pi/2$}$
|
---|
$$
\begin{array}{cc}
\sin(\theta):+ & \csc(\theta):+ \\
\cos(\theta):- & \sec(\theta):- \\
\tan(\theta):- & \cot(\theta):-
\end{array}
$$
|
$$
\begin{array}{cc}
\sin(\theta):+ & \csc(\theta):+ \\
\cos(\theta):+ & \sec(\theta):+ \\
\tan(\theta):+ & \cot(\theta):+
\end{array}
$$
|
$\text{III: $\pi/2\lt\theta\lt3⋅\pi/2$}$
|
$\text{IV: $3⋅\pi/2\le\theta\lt 2⋅\pi$}$
|
---|
$$
\begin{array}{cc}
\sin(\theta):- & \csc(\theta):- \\
\cos(\theta):- & \sec(\theta):- \\
\tan(\theta):+ & \cot(\theta):+
\end{array}
$$
|
$$
\begin{array}{cc}
\sin(\theta):- & \csc(\theta):- \\
\cos(\theta):+ & \sec(\theta):+ \\
\tan(\theta):- & \cot(\theta):-
\end{array}
$$
|
Inverse Functions
$$\sin^{-1}(-\theta)=-\sin^{-1}(\theta)\qquad\cos^{-1}(-\theta)=\pi-\cos^{-1}(\theta)$$
$$\csc^{-1}(-\theta)=-\csc^{-1}(\theta)\qquad\sec^{-1}(-\theta)=\pi-\sec^{-1}(\theta)$$
$$\tan^{-1}(-\theta)=-\tan^{-1}(\theta)\qquad\cot^{-1}(-\theta)=\pi-\cot^{-1}(\theta)$$
4.1.4 Cofunctions & Complementary Angle Identities
Cofunctions
$$\sin(\theta)=\frac{1}{\csc(\theta)}
\qquad
\cos(\theta)=\frac{1}{\sec{\theta}}$$
$$\tan(\theta)=\frac{\sin{\theta}}{\cos{\theta}}=\frac{1}{\cot{\theta}}$$
Complementary Angles
$$\sin\bigg(\frac{\pi}{2}\pm\theta\bigg)=\pm\cos(\theta)\qquad\cos\bigg(\frac{\pi}{2}\pm\theta\bigg)=\mp\sin(\theta)$$
$$\csc\bigg(\frac{\pi}{2}\pm\theta\bigg)=\pm\sec(\theta)\qquad\sec\bigg(\frac{\pi}{2}\pm\theta\bigg)=\mp\csc(\theta)$$
$$\tan\bigg(\frac{\pi}{2}\pm\theta\bigg)=\mp\cot(\theta)\qquad\cot\bigg(\frac{\pi}{2}\pm\theta\bigg)=\mp\tan(\theta)$$
Inverse Cofunctions
$$\sin^{-1}(1/\theta)=\csc^{-1}(\theta),\medspace |\theta| \ge 1$$
$$\cos^{-1}(1/\theta)=\sec^{-1}(\theta),\medspace |\theta| \ge 1$$
$$\tan^{-1}(1/\theta)=\cot^{-1}(\theta),\medspace \phantom{|}\theta\phantom{|} \gt 0$$
Inverse Complementary Angles
$$\sin^{-1}(\theta)+\cos^{-1}(\theta)=\pi/2,\medspace |\theta| \le 1$$
$$\csc^{-1}(\theta)+\sec^{-1}(\theta)=\pi/2 \phantom{,\medspace |\theta| \le 1}$$
$$\tan^{-1}(\theta)+\cot^{-1}(\theta)=\pi/2,\medspace |\theta| \ge 1$$
4.1.5 Graphs
Function |
Period |
Domain |
Range |
|
$\sin(\theta)$ |
$2·\pi$ |
$(-∞,∞)$ |
$[-1,1]$ |
 |
$\cos(\theta)$ |
$2·\pi$ |
$(-∞,∞)$ |
$[-1,1]$ |
 |
$\tan(\theta)$ |
$\pi$ |
$(-∞,∞),$
$x\ne(n-\frac{1}{2})·\pi$
$\forall n \isin \Z$ |
$(-∞,∞)$ |
 |
$\cot(\theta)$ |
$\pi$ |
$(-∞,∞),$
$x\ne n·\pi$
$\forall n \isin \Z$ |
$(-∞,∞)$ |
 |
$\sec(\theta)$ |
$2·\pi$ |
$(-∞,∞),$
$x\ne(n-\frac{1}{2})·\pi$
$\forall n \isin \Z$ |
$|y|\ge 1$ |
 |
$\csc(\theta)$ |
$2·\pi$ |
$(-∞,∞),$
$x\ne n·\pi$
$\forall n \isin \Z$ |
$|y|\ge 1$ |
 |
$\sin^{-1}(\theta)$ |
$-$ |
$|x|\le 1$ |
$|y|\le \pi/2$ |
 |
$\cos^{-1}(\theta)$ |
$-$ |
$|x|\le 1$ |
$0\le y\le\pi$ |
 |
$\tan^{-1}(\theta)$ |
$-$ |
$(-∞,∞)$ |
$|y|\le \pi/2$ |
 |
$\cot^{-1}(\theta)$ |
$-$ |
$(-∞,∞)$ |
$0\lt |y|\lt\pi$ |
 |
$\sec^{-1}(\theta)$ |
$-$ |
$|x|\ge 1$ |
$0\le y\le \pi,$
$y\ne\pi/2$ |
 |
$\csc^{-1}(\theta)$ |
$-$ |
$|x|\ge 1$ |
$|y|\le \pi/2,$
$y\ne 0$ |
 |
All images in this section were taken from
Wolfram Alpha and edited