Solve the equation for x if 0 ≤ x < 2π. Give your answers in radians using exact values only. (Enter your answers as a comma-separated list.) 2 cos x + tan x = sec x

Question

Solve the equation for x if 0 ≤ x < 2π. Give your answers in radians using exact values only. (Enter your answers as a comma-separated list.)

2 cos x + tan x = sec x

 

Answer 1

2cos(x) + tan(x) = sec(x)

2cos(x) + sin(x)/cos(x) = 1/cos(x)

2cos²(x) + sin(x) = 1

2(1 - sin²(x)) + sin(x) = 1

2 - 2sin²(x) + sin(x) = 1

2sin²(x) - sin(x) - 1 = 0

2sin²(x) - 2sin(x) + sin(x) - 1 = 0

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

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

So, sin(x) = 1, -1/2

For 0 ≤ x < 2π, we have that sin(x) = 1 occurs for x = π/2 and sin(x) = -1/2 occurs for x = 7π/6, 11π/6. However, the original equation is not well defined for x = π/2, because tan(π/2) and sec(π/2) are undefined.

So, the solution is

x = 7π/6, 11π/6