Find all solutions to the equation.
$3\(\sin\)\(\theta\)-6=-9$

• A. $\(\theta\)=\(\frac{\pi}{2}\)+2\(\pi\) n$
• B. $\(\theta\)=2\(\pi\) n$
• C. $\(\theta\)=\(\pi\)+2\(\pi\) n$
• D. $\(\theta\)=\(\frac{3\pi}{2}\)+2\(\pi\) n$

Answer: (D)

Verified step by step guidance

1. Start by isolating the sine function in the equation 3sin(θ) - 6 = -9. Add 6 to both sides to get 3sin(θ) = -3.
2. Divide both sides of the equation by 3 to solve for sin(θ), resulting in sin(θ) = -1.
3. Recall that the sine function equals -1 at specific angles. The general solution for sin(θ) = -1 is θ = 3π/2 + 2πn, where n is an integer.
4. Verify the solution by substituting θ = 3π/2 + 2πn back into the original equation to ensure it satisfies 3sin(θ) - 6 = -9.
5. Conclude that the solutions to the equation are given by θ = 3π/2 + 2πn, where n is any integer, representing all angles coterminal with 3π/2.