Question

How do you solve `1+secx=2cosx`?

Answer 1

`0, (2pi)/3, (4pi)/3, 2pi`

Explanation:

1 + sec x = 2cos x
`1 + 1/(cos x) = 2cos x`
`cos x + 1 = 2cos^2 x`
`2cos^2 x - cos x - 1 = 0`
Solve this quadratic equation for cos x.
Use trig table and unit circle.
Since a + b + c = 0, use shortcut. There are 2 real roots:
cos x = 1 and cos x = c/a = - 1/2.
a. cos x = 1 --> x = 0 and `x = 2pi`
b. `cos x = - 1/2` -->arc `x = +- (2pi)/3`
Co-terminal of arc `(-2pi)/3` is arc `(4pi)/3`
Answers for `(0, 2pi)`:
`0, (2pi)/3, (4pi)/3, 2pi`
For general answers add `2kpi`