Question

what is sin \frac{5\pi}{4}?

Explanation:

Step1: Determine the reference angle

The angle \(\frac{5\pi}{4}\) is in the third quadrant. The reference angle \(\theta_{r}\) for an angle \(\theta=\frac{5\pi}{4}\) is \(\theta_{r}=\frac{5\pi}{4}-\pi=\frac{\pi}{4}\).

Step2: Use the sine function value for the reference angle and sign

We know that \(\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}\). In the third quadrant, the sine function is negative. So \(\sin\frac{5\pi}{4}=-\sin\frac{\pi}{4}\).

Answer:

\(-\frac{\sqrt{2}}{2}\), so the number in the green box is \(2\) and the number in the white box is \(2\)