Question

find sin(t) and cos(t) for the values of t whose terminal points are shown on the unit circle in the figure. t increases in increments of π/6.

t

sin(t)

cos(t)

|0|
|π/6|
|π/3|
|π/2|
|2π/3|
|5π/6|

Explanation:

Step1: Recall unit - circle definitions

On the unit circle $x = \cos(t)$ and $y=\sin(t)$.

Step2: Evaluate for $t = 0$

For $t = 0$, the terminal - point is $(1,0)$. So $\sin(0)=0$ and $\cos(0)=1$.

Step3: Evaluate for $t=\frac{\pi}{6}$

For $t=\frac{\pi}{6}$, the terminal - point is $(\frac{\sqrt{3}}{2},\frac{1}{2})$. So $\sin(\frac{\pi}{6})=\frac{1}{2}$ and $\cos(\frac{\pi}{6})=\frac{\sqrt{3}}{2}$.

Step4: Evaluate for $t=\frac{\pi}{3}$

For $t = \frac{\pi}{3}$, the terminal - point is $(\frac{1}{2},\frac{\sqrt{3}}{2})$. So $\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}$ and $\cos(\frac{\pi}{3})=\frac{1}{2}$.

Step5: Evaluate for $t=\frac{\pi}{2}$

For $t=\frac{\pi}{2}$, the terminal - point is $(0,1)$. So $\sin(\frac{\pi}{2}) = 1$ and $\cos(\frac{\pi}{2})=0$.

Step6: Evaluate for $t=\frac{2\pi}{3}$

For $t=\frac{2\pi}{3}$, the terminal - point is $(-\frac{1}{2},\frac{\sqrt{3}}{2})$. So $\sin(\frac{2\pi}{3})=\frac{\sqrt{3}}{2}$ and $\cos(\frac{2\pi}{3})=-\frac{1}{2}$.

Step7: Evaluate for $t=\frac{5\pi}{6}$

For $t=\frac{5\pi}{6}$, the terminal - point is $(-\frac{\sqrt{3}}{2},\frac{1}{2})$. So $\sin(\frac{5\pi}{6})=\frac{1}{2}$ and $\cos(\frac{5\pi}{6})=-\frac{\sqrt{3}}{2}$.

Answer:

$t$	$\sin(t)$	$\cos(t)$
$\frac{\pi}{6}$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$
$\frac{\pi}{3}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$
$\frac{\pi}{2}$	$1$	$0$
$\frac{2\pi}{3}$	$\frac{\sqrt{3}}{2}$	$-\frac{1}{2}$
$\frac{5\pi}{6}$	$\frac{1}{2}$	$-\frac{\sqrt{3}}{2}$