# Thread: Find all values of x in the interval

1. ## Find all values of x in the interval

Find all values of x in the interval [0,2π] that satisfy sin 2x= -sin x

2. Hello, jpenny32!

We need this double-angle identity: . $\sin2\theta \:=\:2\sin\theta\cos\theta$

Find all values of $x$ in the interval $[0,\,2\pi]$ that satisfy: . $\sin 2x\:=\: -\sin x$
Using the identity, we have: . $2\sin x\cos x \:=\:-\sin x \quad\Rightarrow\quad 2\sin x\cos x + \sin x \:=\:0$

Factor: . $\sin x(2\cos x + 1) \:=\:0$

Then: . $\begin{array}{ccccccccccc}\sin x \:=\: 0 & \Rightarrow & \boxed{x \:=\: 0,\,\pi,\,2\pi} \\ \\[-3mm]
2\cos x+1 \:=\: 0 & \Rightarrow & \cos x \:=\: -\frac{1}{2} & \Rightarrow & \boxed{x \:=\:\frac{2\pi}{3},\:\frac{4\pi}{3}} \end{array}$