Solve for all solutions

Question

solve 7cos(2x)=7cos^2(x)-5 for all solutions 0<=x<2pi

x=_________

Give your answers accurate to 2 decimal places 

Answer 1

There is an identity related to cos(2x) that may be useful.

Answer 2

7cos(2x) = 7cos²x - 5

 

7(2cos²x - 1) = 7cos²x - 5

 

7cos²x = 2

 

  cos²x = 2/7

 

  cosx = ±√(2/7) = ±0.5345

 

If cosx = 0.5345, then x = Cos^-1(0.5345) or 2π - Cos^-1(0.5345) = 1.01, 5.28

 

If cosx = -0.5345, then x = π - Cos^-1(0.5345) or π + Cos^-1(0.5345) = 2.13, 4.15