Question

Solve:4cosθ-3secθ=2tanθ?

Answer 1

`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^@`