enter the coordinates of the point on the unit circle at the given angle. 315° (√? / 2, -√2 / )

Explanation:

Step1: Recall the unit - circle coordinates formula

For a point ((x,y)) on the unit circle at an angle (\theta), (x = \cos\theta) and (y=\sin\theta). Given (\theta = 315^{\circ}), (\cos315^{\circ}=\cos(360^{\circ}- 45^{\circ})). Using the identity (\cos(A - B)=\cos A\cos B+\sin A\sin B) with (A = 360^{\circ}), (B = 45^{\circ}), (\cos360^{\circ}=1), (\sin360^{\circ}=0), (\cos45^{\circ}=\frac{\sqrt{2}}{2}), (\sin45^{\circ}=\frac{\sqrt{2}}{2}), we get (\cos315^{\circ}=\cos45^{\circ}=\frac{\sqrt{2}}{2}). So the number under the first square - root is (2).

Step2: Find the (y) - coordinate denominator

For (y=\sin315^{\circ}), (\sin315^{\circ}=\sin(360^{\circ}-45^{\circ})). Using the identity (\sin(A - B)=\sin A\cos B-\cos A\sin B) with (A = 360^{\circ}), (B = 45^{\circ}), (\sin360^{\circ}=0), (\cos360^{\circ}=1), (\sin45^{\circ}=\frac{\sqrt{2}}{2}), (\cos45^{\circ}=\frac{\sqrt{2}}{2}), we get (\sin315^{\circ}=-\sin45^{\circ}=-\frac{\sqrt{2}}{2}). The denominator of the (y) - coordinate is (2).

Answer:

The number in the green box is (2) and the number in the gray box is (2).