Question

How do you solve `sin 4x + sin 2x = 0` using the product and sum formulas?

Answer 1

`\Rightarrow\sin 2(2x) + \sin 2x=0`
`\Rightarrow 2\sin 2x\cos 2x + \sin 2x = 0`
`\Rightarrow \sin 2x(2\cos 2x + 1)=0`

Case 1: `\sin 2x = 0`,
`\Rightarrow 2x = 0\Rightarrow x=0, 2\pi, 4\pi,...`

Case 2: `(2\cos 2x + 1) = 0`
`\Rightarrow 2\cos 2x = -1 \Rightarrow \cos 2x = -\frac{1}{2}`
`\Rightarrow 2x=\frac{2\pi}{3}, \frac{4\pi}{3}, \frac{8\pi}{3},\frac{10\pi}{3},...`
`\Rightarrow x=\frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3},\frac{5\pi}{3},...`

Cosine is positive in the 1st and 4th quadrants and negative in the 2nd and 3rd quadrants, hence the choice of angles.