Question

Question #00e9a

Answer 1

`0; pi/3; (2pi)/3; and (4pi)/3`

Explanation:

Apply the trig identity:
`sin a + sin b = 2sin ((a +b)/2)cos ((a - b)/2)`
`sin 2x + sin 4x = 2sin ((6x)/2)cos ((2x)/2) = 2sin 3x.cos x`
`(sin 2x + sin 4x) + sin 3x = sin 3x(2cos x + 1) = 0`
Either one of the 2 factors must be zero.
a. sin 3x = 0 .
Trig table give 3 solution arcs:
3x = 0 --> x = 0
`3x = pi` --> `x = pi/3`
`3x = 2pi` --> `x = (2pi)/3`
b. 2cos x + 1 = 0 --> `cos x = - 1/2`
Trig table and unit circle give 2 solution arcs:
`x = 2pi/3` and `x = - (2pi)/3` --> or `x = (4pi)/3` (co-terminal)
Answers for `(0, 2pi)`:
`0, pi/3, (2pi)/3, (4pi/3)`