trigonometry help

Question

Algebraically find the solutions of sin(2θ)-cosθ=0 on the interval 0 less than or equal to θ less than or equal to 2pi.

Answer 1

We can use the double angle formula to begin the simplication process...

sin(2θ) = 2*sin(θ)*cos(θ). with 0 <= θ <= 2π

Therefore...

2*sin(θ)*cos(θ) - cos(θ) = 0

Now let x = cos(θ) which leads to sqrt(1-x²) = sin(θ)

2*sqrt(1-x²)*x - x = 0

Bring over the x to the other side of the equation and square both sides to remove the square-root...

4*(1-x²)*x² = x²

With some simplification...

4*(1-x²) = 1

-4x² = -3

x² = 3/4

Now solve for x...

x = +/- sqrt(3)/2

Since x = cos(θ) we want to find the angles which generate x = +/-sqrt(3)/2

θ = π/6, 5π/6, 7π/6, 11π/6

Solution is based on a simple substitution track to change the problem into an algebraic one

Answer 2