Question

How do you find all solutions of the equation in the interval (0, 2pi) `1/2 = 1 - cos^2 x`?

Answer 1

`pi/4, (3pi)/4, (5pi)/4, (7pi)/4`

Explanation:

`1/2 = 1 - cos^2 x`. Replace `(1 - cos^2 x)` by `sin^2 x.`
`1/2 = sin^2 x`
`sin x = +- sqrt2/2`
a. `sin x = sqrt2/2` --> `x = pi/4` and `x = (3pi)/4`
b. `sin x = - sqrt2/2` --> `x = - pi/4` and `x = (-3pi)/4`

Note: `(-pi/4)` --> `(7pi)/4` (co-terminal) and `- (3pi)/4` --> `(5pi)/4` (co-terminal)

Answers for `(0, 2pi)`:
`pi/4, (3pi)/4, (5pi)/4, (7pi)/4`