Question

How do you find all `sin6x+sin2x=0` in the interval `[0,2pi)`?

Answer 1

`0, pi/4, pi/2, (3pi)/4, 2pi`

Explanation:

Use trig identity:
`sin a + sin b = 2sin ((a +b)/2)cos ((a - b)/2)`
In this case -->
f(x) = sin 6x + sin 2x = 2sin (4x).cos (2x) = 0

a. sin 4x = 0
Unit circle gives 3 solutions:
4x = 0 --> x = 0
`4x = pi` --> `x = pi/4`
`4x = 2pi` -->`x = (2pi)/4 = pi/2`
b. cos 2x = 0
trig unit circle gives 2 solutions:
`2x = pi/2` --> `x = pi/4`
`2x = (3pi)/2` --> `x = (3pi)/4`

All answers for `(0, 2pi)`
`0, pi/4, pi/2, (3pi)/4, 2pi`