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) please select the best answer from the choices provided

Explanation:

Step1: Recall unit - circle 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: Rewrite the angle

$-105^{\circ}=-180^{\circ}+75^{\circ}$. So, $\cos(-105^{\circ})=\cos(180^{\circ} - 75^{\circ})=-\cos75^{\circ}$ and $\sin(-105^{\circ})=-\sin75^{\circ}$.

Step3: Use angle - addition formulas

$\cos75^{\circ}=\cos(45^{\circ}+30^{\circ})=\cos45^{\circ}\cos30^{\circ}-\sin45^{\circ}\sin30^{\circ}=\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}-\frac{\sqrt{2}}{2}\times\frac{1}{2}=\frac{\sqrt{6}-\sqrt{2}}{4}\approx0.26$.
$\sin75^{\circ}=\sin(45^{\circ}+30^{\circ})=\sin45^{\circ}\cos30^{\circ}+\cos45^{\circ}\sin30^{\circ}=\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\times\frac{1}{2}=\frac{\sqrt{6}+\sqrt{2}}{4}\approx0.97$.
So, $\cos(-105^{\circ})\approx - 0.26$ and $\sin(-105^{\circ})\approx - 0.97$.

Answer:

C. (-0.26, -0.97)