Question

Find all solutions for the following equations sin(2x)+sin(π/2-x)=0 ?

Answer 1

`pi/2; (3pi)/2 ; (7pi)/6; (11pi)/6` `(+ 2kpi)`

Explanation:

`sin 2x + sin (pi/2 - x) = 0`
Note. sin 2x = 2sin x.cos x
`sin (pi/2 - x) = cos x`
Therefor,
`2sin x.cos x + cos x = 0`
`cos x(2sin x + 1) = 0`
Either factor should be zero -->
a. `cos x = 0`
Unit circle gives -->
`x = pi/2 + 2kpi`, and
`x = (3pi)/2 + 2kpi`
b. `2sin x + 1 = 0` --> `sin x = - 1/2`
Unit circle gives -->
`x = (7pi)/6 + 2kpi`, and
`x = (11pi)/6 + 2kpi`