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

1 Answer
Feb 10, 2018

theta = pi/6 + 2pinθ=π6+2πn , pi/2 + 2pinπ2+2πn

Explanation:

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

cos(2theta) = 1 - 2sin^2(theta)cos(2θ)=12sin2(θ)

Then replacing this into our equation:

1-2sin^2(theta) + 3sin(theta) = 212sin2(θ)+3sin(θ)=2

Then combining like-terms:

-2sin^(theta) + 3sin(theta) - 1 = 02sinθ+3sin(θ)1=0

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

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

So now we have our two zeros of the equation:

-2sin(theta) + 1 = 02sin(θ)+1=0 and sin(theta)-1=0sin(θ)1=0

Simplifying this we get:

sin(theta) = 1/2, 1sin(θ)=12,1

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

Thus we get the answer:

theta = pi/6 + 2pin, pi/2 + 2pinθ=π6+2πn,π2+2πn