Question

Find all solutions to the equation

in the interval [0, 27). Enter the
solutions in increasing order.
cos 2x + cos x = 0

Answer 1

It looks like the equation is

cos(2x) + cos(x) = 0

Use the double angle identity for cosine to write this as

(2 cos²(x) - 1) + cos(x) = 0

2 cos²(x) + cos(x) - 1 = 0

Factorize the left side:

(2 cos(x) - 1) (cos(x) + 1) = 0

Then either

2 cos(x) - 1 = 0 or cos(x) + 1 = 0

2 cos(x) = 1 or cos(x) = -1

cos(x) = 1/2 or cos(x) = -1

[x = cos⁻¹(1/2) + 2nπ or x = -cos⁻¹(1/2) + 2nπ]

… or [x = cos⁻¹(-1) + 2nπ]

(where n is any integer)

x = π/3 + 2nπ or x = -π/3 + 2nπ or x = π + 2nπ

We get the following solutions in the interval [0, 2π):

x = π/3 and x = π and x = 5π/3