Solve:4cosθ-3secθ=2tanθ?

1 Answer
Jun 22, 2018

x = 5^@74 + k360^@
x = 174^@26 + k360^@
x = 216^@87 + k360^@
x = 323^@13+ k360^@)

Explanation:

4cos t - 3/(cos t) = (2sin t)/(cos t)
Multiply both sides by cos t:
4cos^2 t - 3 = 2sin t
4 - 4sin^2 t - 3 = 2sin t
4sin^2 t + 2sin t - 1 = 0
D = d^2 = b^2 - 4ac = 4 + 4 = 8 --> d= +- 2sqrt2
There are 2 real roots:
sin x = -b/(2a) +- d/(2a) = - 2/8 +- (2sqrt2)/8 = (-1 +- sqrt2)/4
sin x = (-1 + sqrt2)/4 = 0.10
sin x = (- 1 - sqrt2)/4 = - 0.60
a. sin x = 0.10
Calculator and unit circle give 2 solutions for x:
x = 5^@74, and x = 180 - 5.74 = 174^@26
b. sin x = -0.60
Calculator and unit circle -->
x = - 36^@87, and x = 180 - (-36.87) = 216^@87
Note x = - 36.87 is co-terminal to x = 323^@13
For general answers, add k360^@