determine the coordinates of the point on the unit circle corresponding to the given central angle. if…

Answer:

A. (-0.97, -0.26)

Explanation:

Step1: Recall the unit - circle coordinate formula

For a central angle (\theta), the coordinates of the point on the unit circle are ((\cos\theta,\sin\theta)).

Step2: Convert the angle to a positive equivalent angle

Since (- 105^{\circ}+360^{\circ}=255^{\circ}), and (\cos(-105^{\circ})=\cos(255^{\circ})), (\sin(-105^{\circ})=\sin(255^{\circ})).

Step3: Calculate (\cos(-105^{\circ}))

(\cos(-105^{\circ})=\cos(180^{\circ} + 75^{\circ})=-\cos(75^{\circ})). Using the formula (\cos(A + B)=\cos A\cos B-\sin A\sin B) with (A = 45^{\circ}), (B = 30^{\circ}), (\cos(75^{\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), so (\cos(-105^{\circ})\approx - 0.97).

Step4: Calculate (\sin(-105^{\circ}))

(\sin(-105^{\circ})=-\sin(105^{\circ})=-\sin(60^{\circ}+45^{\circ})). Using the formula (\sin(A + B)=\sin A\cos B+\cos A\sin B) with (A = 60^{\circ}), (B = 45^{\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}\approx0.97), so (\sin(-105^{\circ})\approx - 0.26).

So the coordinates of the point on the unit circle corresponding to (-105^{\circ}) are ((-0.97,-0.26)).