Question

How do you find the exact value of `3cos2theta-4cos^2theta+2=0` in the interval `0<=theta<360`?

Answer 1

Our goal here is to convert everything to one trigonometric function. Sine would probably be easiest. We hence apply the following identities:

•`sin^2beta + cos^2beta = 1 -> cos^2beta = 1- sin^2beta`
•`cos2beta = 1 - 2sin^2beta`

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`3(1- 2sin^2theta) - 4(1 - sin^2theta) + 2 = 0`

`3 - 6sin^2theta - 4 + 4sin^2theta + 2 = 0`

`1 = 2sin^2theta`

`1/2 = sin^2theta`

`+-1/sqrt(2) = sintheta`

`theta = 45˚, 135˚, 225˚, 315˚`

Hopefully this helps!