Question

identify the points on the unit circle shown above.
a: (1, 0)
b: (\frac{\sqrt{6}+\sqrt{2}}{4}, \frac{\sqrt{6}-\sqrt{2}}{4})
c: (\frac{\sqrt{3}}{2}, \frac{1}{2})
d: (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})
e: (0, 1)
f: (\frac{\sqrt{6}-\sqrt{2}}{4}, \frac{\sqrt{6}+\sqrt{2}}{4})
g: (\frac{1}{2}, \frac{\sqrt{3}}{2})
h: (-\frac{1}{2}, \frac{\sqrt{3}}{2})
i: (-1, 0)

Explanation:

Step1: Recall unit - circle coordinates

On the unit circle \(x = \cos\theta\) and \(y=\sin\theta\). The circle is divided into 24 equal parts. Each part has an angle of \(\frac{360^{\circ}}{24}=15^{\circ}\).

Step2: Calculate coordinates for point B

For point B, the angle \(\theta = 30^{\circ}\). Using the formulas \(x=\cos30^{\circ}=\frac{\sqrt{3}}{2}\) and \(y = \sin30^{\circ}=\frac{1}{2}\).

Step3: Calculate coordinates for point C

For point C, the angle \(\theta = 45^{\circ}\). Using the formulas \(x=\cos45^{\circ}=\frac{\sqrt{2}}{2}\) and \(y=\sin45^{\circ}=\frac{\sqrt{2}}{2}\).

Step4: Calculate coordinates for point F

For point F, the angle \(\theta = 150^{\circ}\). Using the formulas \(x=\cos150^{\circ}=-\frac{\sqrt{3}}{2}\) and \(y=\sin150^{\circ}=\frac{1}{2}\).

Step5: Calculate coordinates for point G

For point G, the angle \(\theta = 165^{\circ}\). Using the sum - of - angles formula \(\cos(A + B)=\cos A\cos B-\sin A\sin B\) and \(\sin(A + B)=\sin A\cos B+\cos A\sin B\) with \(A = 120^{\circ}\), \(B = 45^{\circ}\). \(\cos165^{\circ}=\cos(120^{\circ}+45^{\circ})=\cos120^{\circ}\cos45^{\circ}-\sin120^{\circ}\sin45^{\circ}=(-\frac{1}{2})\times\frac{\sqrt{2}}{2}-\frac{\sqrt{3}}{2}\times\frac{\sqrt{2}}{2}=\frac{-\sqrt{2}-\sqrt{6}}{4}\), \(\sin165^{\circ}=\sin(120^{\circ}+45^{\circ})=\sin120^{\circ}\cos45^{\circ}+\cos120^{\circ}\sin45^{\circ}=\frac{\sqrt{3}}{2}\times\frac{\sqrt{2}}{2}+(-\frac{1}{2})\times\frac{\sqrt{2}}{2}=\frac{\sqrt{6}-\sqrt{2}}{4}\).

Step6: Calculate coordinates for point H

For point H, the angle \(\theta = 180^{\circ}- 30^{\circ}=150^{\circ}\). Using the formulas \(x = \cos150^{\circ}=-\frac{\sqrt{3}}{2}\) and \(y=\sin150^{\circ}=\frac{1}{2}\).

Answer:

B:\((\frac{\sqrt{3}}{2},\frac{1}{2})\)

C:\((\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})\)

F:\((-\frac{\sqrt{3}}{2},\frac{1}{2})\)

G:\((\frac{-\sqrt{2}-\sqrt{6}}{4},\frac{\sqrt{6}-\sqrt{2}}{4})\)

H:\((-\frac{\sqrt{3}}{2},\frac{1}{2})\)