What is the method for solving cosine squared equations?

[member 422733]

I was just finishing my physics homework (don't worry this isn't a HW question, the homework is done) and the last calculation I had to do was cos²(θ) = 0.6. I just plugged this into my calculator to solve for me and got it right.

Now I'm curious though. How would one solve this by hand?

 

[berkeman]

I was just finishing my physics homework (don't worry this isn't a HW question, the homework is done) and the last calculation I had to do was cos²(θ) = 0.6. I just plugged this into my calculator to solve for me and got it right.

Now I'm curious though. How would one solve this by hand?

Take the square root of 0.6 by hand, and use cosine tables?

 

[haruspex]

cos²(θ) = 0.6. I just plugged this into my calculator to solve for me and got it right.
Now I'm curious though. How would one solve this by hand?

If you really mean 'by hand' (no calculator),
cos(2θ) = 2 cos²(θ) -1 = 0.2 = sin(π/2 - 2θ)
π/2 - 2θ ≈ 0.2
θ ≈ π/4 - 0.1
Check (by calculator): cos²(π/4 - 0.1) = 0.5993..