Question

How do you use the quadratic formula to solve `7cos^2theta-1=5costheta` for `0<=theta<360`?

Answer 1

`28^@70; 99^@37; 260^@63; 313^@30`

Explanation:

`7cos^2t - 5cos t - 1 = 0`.
Solve this quadratic equation for cos t:
`D = = b^2 - 4ac = 25 + 28 = 53` --> `d = +- sqrt53`
There are 2 real roots:
`cos t = -b/(2a) +- d/(2a) = 5/14 +- sqrt53/14 = (5 +- sqrt53)/14`
a. `cos t = (5 + sqrt53)/14 = 0.877`
Calculator and trig unit circle give:
`t = +- 28^@70`
b. `cos t = (5 - sqrt53)/14 = - 0.163`
`t = +- 99^@37`
Co-terminal arc of `(-28^@70) --> 313^@30`
Co-terminal arc of`(- 99^@37) --> 260^@63`
Answers for (0, 360)
`28^@70; 99^@37; 260^@63; 313^@30`