Question

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 \(2\).