For sin 2(x) +cos(x)=0, use a double-angle or half-angle formula to simplify the equation and then find all solutions of the equation in the interval [0,2pi)

Question

x₁₌

x₂=

x₃=

x₄=

with x₁<x₂<x₃<x_4.

_{What is the procedure I need to do with the formula?}

Answer 1

Use the double angle identity sin(2x) = 2sin(x)cos(x) and substitute yielding:

2sin(x)cos(x) + cos(x) = 0 and then factor out cos(x) to get:

cos(x)(2sin(x) + 1) = 0 then set each piece equal to zero (using the zero product rule)

cos(x) = 0 when x = π/2 and 3π/2

2sin(x) + 1 = 0 or sin(x) = -1/2 when x = 7π/6 and 11π/6

You'll need to reorder them to comply with x1<x2<x3<x4