4.1 Contents

• Unit Circle
• Inverse, Reciprocal & Power Notation
• Even/Odd Identities & Reflections
• Cofunctions & Complementary Angle Identities
• Graphs

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
$\sin(\theta)$	$2·\pi$	$(-∞,∞)$	$[-1,1]$
$\cos(\theta)$	$2·\pi$	$(-∞,∞)$	$[-1,1]$
$\tan(\theta)$	$\pi$	$(-∞,∞),$ $x\ne(n-\frac{1}{2})·\pi$ $\forall n \isin \Z$	$(-∞,∞)$
$\cot(\theta)$	$\pi$	$(-∞,∞),$ $x\ne n·\pi$ $\forall n \isin \Z$	$(-∞,∞)$
$\sec(\theta)$	$2·\pi$	$(-∞,∞),$ $x\ne(n-\frac{1}{2})·\pi$ $\forall n \isin \Z$	$|y|\ge 1$
$\csc(\theta)$	$2·\pi$	$(-∞,∞),$ $x\ne n·\pi$ $\forall n \isin \Z$	$|y|\ge 1$
$\sin^{-1}(\theta)$	$-$	$|x|\le 1$	$|y|\le \pi/2$
$\cos^{-1}(\theta)$	$-$	$|x|\le 1$	$0\le y\le\pi$
$\tan^{-1}(\theta)$	$-$	$(-∞,∞)$	$|y|\le \pi/2$
$\cot^{-1}(\theta)$	$-$	$(-∞,∞)$	$0\lt |y|\lt\pi$
$\sec^{-1}(\theta)$	$-$	$|x|\ge 1$	$0\le y\le \pi,$ $y\ne\pi/2$
$\csc^{-1}(\theta)$	$-$	$|x|\ge 1$	$|y|\le \pi/2,$ $y\ne 0$