Question

Solve each equation for all real values of x.

7.) 3 cos 2x - 5 cos x = 1
8.) 2 sin^2 x-5 sin x + 2 = 0
9.)3 sec^2 - 4 = 0
10.) tan x (tan x-1) = 0

Answer 1

7)
since cos 2x = 2cos^2 x - 1, we have
6cos^2 x - 5cos x - 4 = 0
(2cos x + 1)(3cos x - 4) = 0
cos x = -1/2 --> x = 2pi/3 or 5pi/3
cos x = 4/3 --> no solution

8)
(sin x - 2)(2sin x - 1) = 0
sin x = 2 --> no solution
sin x = 1/2 --> x = pi/6 or 5pi/6

9)
3sec^2 x = 4
sec x = ±2/√3
or, more familiarly,
cos x = ±√3/2
x = pi/6, 5pi/6, 7pi/6, 11pi/6

10)
tan X = 0 --> x = 0, pi
tan x = 1 --> x = pi/4, 5pi/4