Solve for the angle

Question

Solve for the angle θ where 0≤ θ ≤ 2∏

sin 2θ + square root of 2 cos θ = 0

θ =

Answer 1

sin 2θ + √2 cos θ = 0 and we are looking for solutions on [0,2π]

Since sin 2θ = 2(sin θ)(cos θ) this means that

2 (sin θ)(cos θ) + √2 cos θ = 0

There is a common factor of cos θ in the two LHS terms, so we factor

cos θ(2sin θ – √2) = 0

If either factor is zero we have a solution to the trigonometric equation.

So our solutions come from cos θ = 0,

which gives us solutions θ = π/2 and 3π/2,

and 2sin θ = √2 which means that sin θ = √2/2

which gives us solutions θ = π/4 and 3π/4

Solutions (in increasing order)

π/4, π/2, 3π/4, 3π/2