Exact values on a interval involving the unit circle.

Question

Find the exact values of all the solutions on the interval [0,2pi) to the equation 10cos(2theta)=cos^theta+6

Answer 1

10 cos (2x) = cos²x + 6. This is the equation I'm assuming you want me to solve.

In order to solve this equation, we need to get everything in terms of cos x, or cos²x. An applicable identity here is cos(2x) = 2cos²x - 1.

This means we can rewrite our original equation as 10(2cos²x - 1) = cos²x + 6.

20 cos²x - 10 = cos²x + 6.

19 cos²x - 16 = 0.

19 cos²x = 16

cos²x = 16/19

cos x = 4/√19

The approximate solutions (I had to look this up) are 3π/23 and 20π/23.