use the unit circle to find the exact value of the sin, cos, tan, csc, sec, and cot of each angle. 5) 135°…

Explanation:

Step1: Recall unit - circle definitions

On the unit circle, for an angle $\theta$, $\sin\theta = y$, $\cos\theta=x$, $\tan\theta=\frac{y}{x}(x\neq0)$, $\csc\theta=\frac{1}{y}(y\neq0)$, $\sec\theta=\frac{1}{x}(x\neq0)$, $\cot\theta=\frac{x}{y}(y\neq0)$.

Step2: Convert $135^{\circ}$ to radians

$135^{\circ}=135\times\frac{\pi}{180}=\frac{3\pi}{4}$ radians. The coordinates of the point on the unit - circle corresponding to $\theta = \frac{3\pi}{4}$ are $(-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})$. $\sin(135^{\circ})=\sin(\frac{3\pi}{4})=\frac{\sqrt{2}}{2}$ $\cos(135^{\circ})=\cos(\frac{3\pi}{4})=-\frac{\sqrt{2}}{2}$ $\tan(135^{\circ})=\tan(\frac{3\pi}{4})=\frac{\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}}=- 1$ $\csc(135^{\circ})=\frac{1}{\sin(135^{\circ})}=\sqrt{2}$ $\sec(135^{\circ})=\frac{1}{\cos(135^{\circ})}=-\sqrt{2}$ $\cot(135^{\circ})=\frac{1}{\tan(135^{\circ})}=-1$

Step3: Convert $-120^{\circ}$ to radians

$-120^{\circ}=-120\times\frac{\pi}{180}=-\frac{2\pi}{3}$ radians. The coordinates of the point on the unit - circle corresponding to $\theta=-\frac{2\pi}{3}$ are $(-\frac{1}{2},-\frac{\sqrt{3}}{2})$. $\sin(-120^{\circ})=\sin(-\frac{2\pi}{3})=-\frac{\sqrt{3}}{2}$ $\cos(-120^{\circ})=\cos(-\frac{2\pi}{3})=-\frac{1}{2}$ $\tan(-120^{\circ})=\tan(-\frac{2\pi}{3})=\frac{-\frac{\sqrt{3}}{2}}{-\frac{1}{2}}=\sqrt{3}$ $\csc(-120^{\circ})=\frac{1}{\sin(-120^{\circ})}=-\frac{2\sqrt{3}}{3}$ $\sec(-120^{\circ})=\frac{1}{\cos(-120^{\circ})}=-2$ $\cot(-120^{\circ})=\frac{1}{\tan(-120^{\circ})}=\frac{\sqrt{3}}{3}$

Step4: Analyze $\theta =-\frac{5\pi}{3}$

The coordinates of the point on the unit - circle corresponding to $\theta =-\frac{5\pi}{3}$ are $(\frac{1}{2},\frac{\sqrt{3}}{2})$. $\sin(-\frac{5\pi}{3})=\frac{\sqrt{3}}{2}$ $\cos(-\frac{5\pi}{3})=\frac{1}{2}$ $\tan(-\frac{5\pi}{3})=\frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}=\sqrt{3}$ $\csc(-\frac{5\pi}{3})=\frac{1}{\sin(-\frac{5\pi}{3})}=\frac{2\sqrt{3}}{3}$ $\sec(-\frac{5\pi}{3})=\frac{1}{\cos(-\frac{5\pi}{3})}=2$ $\cot(-\frac{5\pi}{3})=\frac{1}{\tan(-\frac{5\pi}{3})}=\frac{\sqrt{3}}{3}$

Step5: Analyze $\theta=\frac{7\pi}{4}$

The coordinates of the point on the unit - circle corresponding to $\theta=\frac{7\pi}{4}$ are $(\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2})$. $\sin(\frac{7\pi}{4})=-\frac{\sqrt{2}}{2}$ $\cos(\frac{7\pi}{4})=\frac{\sqrt{2}}{2}$ $\tan(\frac{7\pi}{4})=\frac{-\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}=-1$ $\csc(\frac{7\pi}{4})=\frac{1}{\sin(\frac{7\pi}{4})}=-\sqrt{2}$ $\sec(\frac{7\pi}{4})=\frac{1}{\cos(\frac{7\pi}{4})}=\sqrt{2}$ $\cot(\frac{7\pi}{4})=\frac{1}{\tan(\frac{7\pi}{4})}=-1$

Answer:

For $135^{\circ}$: $\sin(135^{\circ})=\frac{\sqrt{2}}{2}$, $\cos(135^{\circ})=-\frac{\sqrt{2}}{2}$, $\tan(135^{\circ})=-1$, $\csc(135^{\circ})=\sqrt{2}$, $\sec(135^{\circ})=-\sqrt{2}$, $\cot(135^{\circ})=-1$ For $-120^{\circ}$: $\sin(-120^{\circ})=-\frac{\sqrt{3}}{2}$, $\cos(-120^{\circ})=-\frac{1}{2}$, $\tan(-120^{\circ})=\sqrt{3}$, $\csc(-120^{\circ})=-\frac{2\sqrt{3}}{3}$, $\sec(-120^{\circ})=-2$, $\cot(-120^{\circ})=\frac{\sqrt{3}}{3}$ For $-\frac{5\pi}{3}$: $\sin(-\frac{5\pi}{3})=\frac{\sqrt{3}}{2}$, $\cos(-\frac{5\pi}{3})=\frac{1}{2}$, $\tan(-\frac{5\pi}{3})=\sqrt{3}$, $\csc(-\frac{5\pi}{3})=\frac{2\sqrt{3}}{3}$, $\sec(-\frac{5\pi}{3})=2$, $\cot(-\frac{5\pi}{3})=\frac{\sqrt{3}}{3}$ For $\frac{7\pi}{4}$: $\sin(\frac{7\pi}{4})=-\frac{\sqrt{2}}{2}$, $\cos(\frac{7\pi}{4})=\frac{\sqrt{2}}{2}$, $\tan(\frac{7\pi}{4})=-1$, $\csc(\frac{7\pi}{4})=-\sqrt{2}$, $\sec(\frac{7\pi}{4})=\sqrt{2}$, $\cot(\frac{7\pi}{4})=-1$