Find all solutions to the equation.3sin⁡θ−6=−93\sin\theta-6=-93sinθ−6=−9

θ = 3 π 2 + 2 π n \theta=\frac{3\pi}{2}+2\pi n θ = 2 3 π + 2 πn

Find all solutions to the equation. 3 sin ⁡ θ − 6 = − 9 3\sin\theta-6=-9 3 sin θ − 6 = − 9

This problem, we're asked to find all solutions to the equation 3 sine theta minus 6 is equal to negative 9. Now we're going to start with step number 1 of how to solve linear trig equations. So we want to go ahead and isolate our trig function using whatever operations I need to. So here, my trig function is the sine of theta. That's what I'm aiming to isolate. So the first step in doing that is going to be to add 6 to both sides. So that will cancel out, leaving me with 3 sine of theta is equal to negative 9 plus 6, which is negative 3. Now from here to isolate that sine theta, I can just divide both sides by 3. Then that 3 will cancel, leaving me with sine of theta is equal to negative 3 divided by positive 3, which is going to give me negative 1. So step number 1 is done here, and we can go ahead and move on to step number 2, which is to find all of our solutions on the unit circle. So we want to find every angle for which the sine is equal to negative one. So coming down here to my unit circle looking for where my sine is equal to negative one, I know that down here at 3 pi over 2, that's where my sine is going to be. That's negative one. So here I get that theta is equal to 3 pi over 2. Now there are no other angles on my unit circle for which that is true, so that represents all of my solutions that are on that unit circle. So I have completed step number 2. Now for step number 3, we are told that if our domain is not restricted, we need to add 2pyn to each solution. So always remember to double check what you're asked to find in the problem. In this case, our problem asked us to find all of the solutions, so that means that our domain is not restricted and we do want to add 2 pi in here. So step number 3 is done, and then step number 4 is to isolate theta. Now theta is already by itself and completely isolated here, so step number 4 is done as well. And we have our final solution, which is 3pitwo+2pitin. That represents all solutions to the given equation. Thanks for watching, and let me know if you have any questions.

Find all solutions to the equation. 3 sin ⁡ θ − 6 = − 9 3\sin\theta-6=-9 3 sin θ − 6 = − 9

θ = 3 π 2 + 2 π n \theta=\frac{3\pi}{2}+2\pi n θ = 2 3 π + 2 πn

θ = π 2 + 2 π n \theta=\frac{\pi}{2}+2\pi n θ = 2 π + 2 πn

θ = 2 π n \theta=2\pi n θ = 2 πn

θ = π + 2 π n \theta=\pi+2\pi n θ = π + 2 πn

11. Inverse Trigonometric Functions and Basic Trig Equations

Linear Trigonometric Equations

Precalculus

11. Inverse Trigonometric Functions and Basic Trig Equations

Linear Trigonometric Equations

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Multiple Choice

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

$\theta=\frac{\pi}{2}+2\pi n$

$\theta=2\pi n$

$\theta=\pi+2\pi n$

$\theta=\frac{3\pi}{2}+2\pi n$

Verified step by step guidance

Start by isolating the sine function in the equation 3sin(θ) - 6 = -9. Add 6 to both sides to get 3sin(θ) = -3.

Divide both sides of the equation by 3 to solve for sin(θ), resulting in sin(θ) = -1.

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.

Verify the solution by substituting θ = 3π/2 + 2πn back into the original equation to ensure it satisfies 3sin(θ) - 6 = -9.

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.

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