Question

How do you solve `sin^2(x) + sinx = 0`, and solve the equation on interval of (0, 2π)?

Answer 1

First transform the equation f(x) = sin^2 x + sin x = 0 to 2 basic trig equations.
Reminder. There are 4 basic trig equations. They are: sin x = a; cos x = a; tan x = a; and cot x = a.
f(x) = sin x*(1 + sin x) = 0
Next, solve the 2 basic trig equations: sin x = 0; and sin x + 1 = 0, using the trig unit circle.
sin x = 0 --> x = 0, and x = Pi; and x = 2Pi
sin x = -1 = sin 3Pi/2.
Answers within period (0, 2Pi) are: 0; Pi; 3Pi/2; 2Pi
Check.
x = Pi --> sin x = 0 --> f(x) = 0 + 0 = 0
x = 3Pi/2 --> sin x = -1 --> f(x) = (-1)^2 + (-1) = 0