Question

How do you find the exact value of `3cos2theta+costheta+2=0` in the interval `0<=theta<2pi`?

Answer 1

`x = +- 120^@`
`x = +- 70^@53`

Explanation:

Replace cos 2t by `(2cos^2 t - 1)` -->
`3(2cos^2 t - 1) + cos t + 2 = 0`
Solve this quadratic equation for cos t
`6cos^2 t + cos t - 1 = 0`
`D = d^2 = b^2 - 4ac = 1 + 24 = 25` --> `d = +- 5`
There are 2 real roots:
`cos x = -b/(2a) +- d/(2a) = - 1/12 +- 5/12 = (-1 +- 5)/12`
`cos x = - 6/12 = -1/2`
`cos x = 4/12 = 1/3`
Use calculator and unit circle:
a. `cos x = - 1/2` --> `x = +- 120^@`
b. `cos x = 1/3` --> `x = +- 70^@53`