A geometry problem by A Former Brilliant Member
Geometry
Level
1
Number of solutions of the equation tan x + sec x = 2 cos x, lying in the interval [0, 2π] is
The answer is 2.
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1 solution
tan x + sec x = 2 cos x
sin x + 1 = 2 cos2 x, cos x ≠ 0
sin x + 1 = 2 (1 - sin2 x)
2 sin2 x + sin x - 1 = 0
This is quadratic equation in sin x. Solve for sin x
sin x = - 1, or sin x = ½
sin x = -1 ⇒ cos x = 0, which is not possible [cos x ≠ 0 from above]
∴ sin x = ½
x = π or x = π - (π ⁄ 6), in the interval [0, 2π]
x = π or x = 5π ⁄ 6