Verified answer

cos(2θ)=456/9025

You can work it out in degrees or radians, depending on your domain.

Let u = 2θ

Cos (u)= 456/9025

U= cos^-1(456/9025)

U= 1.5202 rads + 2npi [ or 87.10 + 360n] degrees

Or in quadrant IV:

U= 4.7630 rads + 2npi [ or 272.90 + 360n]degrees

Now plug in 2θ for u

2θ= 1.5202 or 4.7630 + 2npi

θ= (0.7601 or 2.3815 ) + npi rads

Or

2θ= 87.10 or 272.90 + 360n

θ= (43.6 or 136.45 )+ 180n degrees

Now if your domain was just [0, 2pi), add pi to the two answers

If the domain was [0,360) add 180 to the two answers.

Hoping this helps!

Edit: Adrian's method would work also, but when you take the square root, put +/ - to get all possible answers.

The multiple angle method above will also work if the angle is 3x, or 5x, etc, so it is good to know.

OK. cos(a+b) = cos(a)*cos(b)-sin(a)*sin(b). a = b so you have cos(2a) = cos^2(a)-sin^2(a)

And this = 1-2sin^2(a)

So, let a = theta and then set 1-2sin^2(theta) = 456/9025.

sin^2(theta) = 0.5- 456/18050, so find the difference and then take the square root. Then take the inverse sin to get theta. Should be somewhere around 43.552 degrees in Q1 and cos is + in Q1 and Q4, so you'll have another answer. 360-43.552 = 316.448 degrees.

If you double these angles and take cos they'll both = 0.05053 approx.

Use the double angle formula