Question

9/ sin x = -√2/2

Explanation:

Step1: Recall the unit - circle values

We know that \(\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}\). Since \(\sin x =-\frac{\sqrt{2}}{2}\), we look for angles in the third and fourth quadrants (where sine is negative).

Step2: Find the angles in the third and fourth quadrants

In the third quadrant, \(x=\pi+\frac{\pi}{4}=\frac{5\pi}{4}+ 2k\pi\), \(k\in\mathbb{Z}\). In the fourth quadrant, \(x = 2\pi-\frac{\pi}{4}=\frac{7\pi}{4}+2k\pi\), \(k\in\mathbb{Z}\).

Answer:

\(x=\frac{5\pi}{4}+2k\pi\) or \(x=\frac{7\pi}{4}+2k\pi\), \(k\in\mathbb{Z}\)