Question

determine the coordinates of the point on the unit circle corresponding to the given central angle. if necessary, round your results to the nearest hundredth. -105° a (-0.97, -0.26) b (1, -0.97) c (-0.26, -0.97) d (-0.26, 0)

Explanation:

Step1: Recall unit - circle coordinate formula

For a point $(x,y)$ on the unit circle corresponding to an angle $\theta$, $x = \cos\theta$ and $y=\sin\theta$. Here $\theta=- 105^{\circ}$.

Step2: Use angle - addition formula

We know that $\cos(-105^{\circ})=\cos(105^{\circ})=\cos(60^{\circ} + 45^{\circ})=\cos60^{\circ}\cos45^{\circ}-\sin60^{\circ}\sin45^{\circ}=\frac{1}{2}\times\frac{\sqrt{2}}{2}-\frac{\sqrt{3}}{2}\times\frac{\sqrt{2}}{2}=\frac{\sqrt{2}-\sqrt{6}}{4}\approx - 0.26$.

Step3: Calculate sine value

$\sin(-105^{\circ})=-\sin(105^{\circ})=-\sin(60^{\circ}+45^{\circ})=-(\sin60^{\circ}\cos45^{\circ}+\cos60^{\circ}\sin45^{\circ})=-(\frac{\sqrt{3}}{2}\times\frac{\sqrt{2}}{2}+\frac{1}{2}\times\frac{\sqrt{2}}{2})=\frac{-\sqrt{6}-\sqrt{2}}{4}\approx - 0.97$.

Answer:

C. (-0.26, -0.97)