Question

How do you solve `cos 2x = - sqrt3 / 2` using the double angle identity?

Answer 1

Solve: `cos 2x = -sqrt3/2`

Ans: `+- 75^@ and +- 105^@`

Explanation:

Solving by double angle identity: `cos 2a = 2cos^2 a - 1`
`2cos^2 x - 1 = - sqrt3/2`
`2cos^2 x = 1 - sqrt3/2 = (2 - sqrt3)/2`
`cos^2 x = (2 - sqrt3)/4`
cos x = +- sqrt(2 - sqrt3)/2 = +-0.517/2 = +- 0.258

a. cos x = 0.258 --> `x = +- 75^@`
b. cos x = - 0.258 --> `x = +- 105^@`
Check by calculator :
`x = +- 75` --> `2x = +- 150` --> cos 2x = - 0.87 = -sqrt3/2. OK
`x = +- 105` --> `2x = +-210` --> cos 2x = - 0.87. OK