Sin2x plus cosx equals 0

Answer 1

sin 2x + cos x = 0 (substitute 2sin x cos x for sin 2x)
2sin x cos x + cos x = 0 (divide by cos x each term to both sides)
2sin x + 1 = 0 (subtract 1 to both sides)
2sin x = -1 (divide by 2 to both sides)
sin x = -1/2

Because the period of the sine function is 360⁰, first find all solutions in [0, 360⁰].


Because sin 30⁰ = 1/2 , the solutions of sin x = -1/2 in [0, 360] are


x = 180⁰ + 30⁰ = 210⁰ (the sine is negative in the third quadrant)
x = 360⁰ - 30⁰ = 330⁰ (the sine is negative in the fourth quadrant)


Thus, the solutions of the equation are given by


x = 210⁰ + 360⁰n and x = 330⁰ + 360⁰n, where n is any integer.