Question

If 0 ≤ x < 2π, then the number of real values of x, which satisfy the equation cos x + cos 2x + cos 3x + cos 4x = 0, is ______

Options

• 5

• 7

• 9

• 3

Answer 1

If 0 ≤ x < 2π, then the number of real values of x, which satisfy the equation cos x + cos 2x + cos 3x + cos 4x = 0, is 7.

Explanation:

cos x + cos 2x + cos 3x + cos 4x = 0

∴ cosx + cos4x + cos2x + cos3x = 0

∴ `2cos  (5x)/2.cos  (3x)/2 + 2cos  (5x)/2.cos  x/2 = 0`

⇒ `cos  (5x)/2.[cos  (3x)/2 + cos  x/2] = 0`

⇒ `cos  (5x)/2 .2cosx.cos  x/2 = 0`

⇒ x = (2n + 1)`pi/5`, (2k + 1)`pi/2` or (2m + 1) π

⇒ `x = pi/5, (3pi)/5, pi, (7pi)/5, (9pi)/5, pi/2, (3pi)/2` in 0 ≤ x < `2pi`