Question

How do you find the general solutions for `sin(2a)-cos(a)-2sin(a)+1=0`?

Answer 1

Solve sin 2x - cos x - 2sin x + 1 = 0

Ans: `2pi; pi/6, and (5pi)/6`

Explanation:

Replace in the equation sin 2x by (2sin x.cos x), we get:
2sin x.cos x - cos x - 2sin x + 1 = 0
Put 2sin x in common factor:
2sin x(cos x - 1) - (cos x - 1) = 0
(cos x - 1)(2sin x - 1) = 0

a. cos x - 1 = 0 --> cos x = 1 --> x = 0 and `x = 2pi`
b. 2sin x - 1 = 0 --> `sin x = 1/2` --> `x = pi/6` and `x = (5pi)/6`
General solution:
`x = 2kpi`
`x = pi/6 + 2kpi`
`x = (5pi)/6 + 2kpi`