Question

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

Answer 1

`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`