Question

How do you solve `Cos 2 theta = cos^2 theta - 1/2`?

Answer 1

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

Explanation:

Replace in the equation cos 2x by `(2cos^2 x - 1)` -->
`2cos^2 x - 1 = cos^2 x - 1/2`
`cos^2 x = 1 - 1/2 = 1/2`
`cos x = +- 1/sqrt2 = +- sqrt2/2`
Trig table and unit circle give -->
a. `cos x = sqrt2/2` --> `x = +- pi/4`
b. `cos x = -sqrt2/2` --> `x = +- (3pi)/4`
Answers for `(0, 2pi)` -->
`pi/4, (3pi)/4, (5pi)/4, (7pi)/4`

Note. Arc `-pi/4` is co-terminal to arc `(7pi)/4`
Arc `(-3pi)/4` is co-terminal to arc `(5pi)/4`