Best way to solve: cos 2 theta + 3 sin theta = 2 ?

1 Answer
Feb 10, 2018

theta = pi/6 + 2pin , pi/2 + 2pin

Explanation:

First we have to use the Double Angle Theorem for cosine:

cos(2theta) = 1 - 2sin^2(theta)

Then replacing this into our equation:

1-2sin^2(theta) + 3sin(theta) = 2

Then combining like-terms:

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

Now we can see that this is simply a quadratic in disguise, to see this easier you can replace sin(theta) with a variable. We can now factor this into:

(-2sin(theta)+1)*(sin(theta)-1)=0

So now we have our two zeros of the equation:

-2sin(theta) + 1 = 0 and sin(theta)-1=0

Simplifying this we get:

sin(theta) = 1/2, 1

But we cannot exclude all the others solutions of the sinusoidal function, so we add by its period times n, an integer.

Thus we get the answer:

theta = pi/6 + 2pin, pi/2 + 2pin