Use trigonometric identities to solve each equation for the

given domain.

sin 2θ = -cos θ for 0 ≤ θ < 2π

*Anyone who has any idea which identity to use is welcome to help.

To solve the equation sin 2θ = -cos θ for the given domain 0 ≤ θ < 2π, we can use trigonometric identities to simplify and find the solutions.

Let's start by using the double-angle identity for sine:

sin 2θ = 2 sin θ cos θ

Now, let's rewrite the equation using this identity:

2 sin θ cos θ = -cos θ

Moving all terms to one side of the equation, we get:

2 sin θ cos θ + cos θ = 0

Now, let's factor out cos θ:

cos θ(2 sin θ + 1) = 0

To find the solutions, we set each term equal to zero:

cos θ = 0
2 sin θ + 1 = 0

For cos θ = 0, we know that the cosine function equals zero at π/2 and 3π/2.
So, we have two solutions for this part: θ = π/2 and θ = 3π/2.

For 2 sin θ + 1 = 0, we solve for sin θ:

2 sin θ = -1
sin θ = -1/2

Using the unit circle or trigonometric functions, we can determine that sin θ equals -1/2 at two points on the unit circle: 7π/6 and 11π/6.

Thus, we have two additional solutions: θ = 7π/6 and θ = 11π/6.

Therefore, combining all the solutions, the equation sin 2θ = -cos θ has four solutions for 0 ≤ θ < 2π: θ = π/2, θ = 3π/2, θ = 7π/6, and θ = 11π/6.