Question

How do you solve `sin2x=-sinx`?

Answer 1

Solve `f(x) = sin 2x + sin x = 0`.
Use the trig identity:`sin 2x = 2.sinx.cos x`

f(x) = sin x(2.cos x - 1) = 0.

Next solve the 2 basic trig equation: `sin x = 0 and 2.cos x = 1.`

`sin x = 0 --> x = 0; and x = 2Pi`

`cos x = 1/2 --> trig conversion table gives -> cos x = cos (Pi/3) --> x = Pi/3`.
Trig unit circle gives another arc that has the same cos value (1/2):
`x = (5Pi)/3.`
`Within interval (0, 2Pi): there are 4 answers: 0; Pi/3; 5Pi/3; and 2Pi`.