Question

name:____ use the triangle and sector pieces to fill in all the blanks on the unit circle. look for patterns and relationships!

Explanation:

Step1: Recall unit - circle coordinates formula

For a point $(x,y)$ on the unit circle $x = \cos\theta$ and $y=\sin\theta$, where $\theta$ is the angle measured counter - clockwise from the positive x - axis.

Step2: Calculate coordinates for $0^{\circ}$ ($\theta = 0$)

$x=\cos(0)=1$, $y = \sin(0)=0$. So the coordinate is $(1,0)$.

Step3: Calculate coordinates for $30^{\circ}$ ($\theta=\frac{\pi}{6}$)

$x=\cos(\frac{\pi}{6})=\frac{\sqrt{3}}{2}$, $y=\sin(\frac{\pi}{6})=\frac{1}{2}$. So the coordinate is $(\frac{\sqrt{3}}{2},\frac{1}{2})$.

Step4: Calculate coordinates for $45^{\circ}$ ($\theta=\frac{\pi}{4}$)

$x=\cos(\frac{\pi}{4})=\frac{\sqrt{2}}{2}$, $y=\sin(\frac{\pi}{4})=\frac{\sqrt{2}}{2}$. So the coordinate is $(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})$.

Step5: Calculate coordinates for $60^{\circ}$ ($\theta=\frac{\pi}{3}$)

$x=\cos(\frac{\pi}{3})=\frac{1}{2}$, $y=\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}$. So the coordinate is $(\frac{1}{2},\frac{\sqrt{3}}{2})$.

Step6: Calculate coordinates for $90^{\circ}$ ($\theta=\frac{\pi}{2}$)

$x=\cos(\frac{\pi}{2}) = 0$, $y=\sin(\frac{\pi}{2})=1$. So the coordinate is $(0,1)$.

Step7: Use symmetry for other quadrants

For example, in the second quadrant, for $120^{\circ}$ ($\theta=\frac{2\pi}{3}$), $x=\cos(\frac{2\pi}{3})=-\frac{1}{2}$, $y=\sin(\frac{2\pi}{3})=\frac{\sqrt{3}}{2}$, coordinate is $(-\frac{1}{2},\frac{\sqrt{3}}{2})$. In the third quadrant, for $210^{\circ}$ ($\theta=\frac{7\pi}{6}$), $x=\cos(\frac{7\pi}{6})=-\frac{\sqrt{3}}{2}$, $y=\sin(\frac{7\pi}{6})=-\frac{1}{2}$, coordinate is $(-\frac{\sqrt{3}}{2},-\frac{1}{2})$. In the fourth quadrant, for $300^{\circ}$ ($\theta=\frac{5\pi}{3}$), $x=\cos(\frac{5\pi}{3})=\frac{1}{2}$, $y=\sin(\frac{5\pi}{3})=-\frac{\sqrt{3}}{2}$, coordinate is $(\frac{1}{2},-\frac{\sqrt{3}}{2})$.

Answer:

$(0^{\circ}):(1,0)$; $(30^{\circ}):(\frac{\sqrt{3}}{2},\frac{1}{2})$; $(45^{\circ}):(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})$; $(60^{\circ}):(\frac{1}{2},\frac{\sqrt{3}}{2})$; $(90^{\circ}):(0,1)$; $(120^{\circ}):(-\frac{1}{2},\frac{\sqrt{3}}{2})$; $(135^{\circ}):(-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})$; $(150^{\circ}):(-\frac{\sqrt{3}}{2},\frac{1}{2})$; $(180^{\circ}):(-1,0)$; $(210^{\circ}):(-\frac{\sqrt{3}}{2},-\frac{1}{2})$; $(225^{\circ}):(-\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2})$; $(240^{\circ}):(-\frac{1}{2},-\frac{\sqrt{3}}{2})$; $(270^{\circ}):(0, - 1)$; $(300^{\circ}):(\frac{1}{2},-\frac{\sqrt{3}}{2})$; $(315^{\circ}):(\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2})$; $(330^{\circ}):(\frac{\sqrt{3}}{2},-\frac{1}{2})$