Question

Use the given information to find the exact value of sin(θ), where sin(θ2)sin(θ) = -2√65 and 3π/2 < θ < 2π.

a) 2/√65
b) -4/√65
c) 1/√65
d) -2/√65

Answer 1

To find the exact value of sin(θ), use the given equation and the Pythagorean identity to solve for sin(θ) and cos(θ). After solving a quadratic equation, you can find that sin(θ) = 2/√65.

Step-by-step explanation:

To find the exact value of sin(θ), we need to solve the equation sin(θ^2)sin(θ) = -2√65, where 3π/2 < θ < 2π.

We can rewrite the equation as sin(θ^2) = -2√65/sin(θ).

Using the identity sin(a + b) = sin(a)cos(b) + cos(a)sin(b), we have sin(θ^2) = sin(θ)cos(θ).

Substituting this identity into the equation, we get sin(θ)cos(θ) = -2√65/sin(θ).

Multiplying both sides by sin(θ), we have sin^2(θ) = -2√65cos(θ).

Now we can use the Pythagorean identity sin^2(θ) + cos^2(θ) = 1 to solve for sin(θ).

Since sin^2(θ) = 1 - cos^2(θ), we can substitute this into the equation to get 1 - cos^2(θ) = -2√65cos(θ).

Expanding and rearranging the equation, we have cos^2(θ) + 2√65cos(θ) - 1 = 0.

Using the quadratic formula, we can solve for cos(θ) and then find sin(θ) using the Pythagorean identity.

After solving the quadratic equation, we find that cos(θ) = -4√65/(√65)^2 = -4/√65.

Substituting this value back into the Pythagorean identity, we find that sin(θ) = √(1 - (-4/√65)^2) = 2/√65.

Therefore, the exact value of sin(θ) is 2/√65.