Solve the following trigonometric equation assuming that 0 ≤ 𝜃 < 2𝜋. (Enter your answers as a comma-separated list.) 3 − 6 sin(𝜃) = 4 − 8 sin(𝜃)

Question

Solve the following trigonometric equation assuming that 0 ≤ 𝜃 < 2𝜋.

 (Enter your answers as a comma-separated list.)

3 − 6 sin(𝜃) = 4 − 8 sin(𝜃)

Why is 𝜃= pi/6 + 2pin and 𝜃 5pi/6 + 2pin wrong and how do I get the correct answer?

Answer 1

Hi Rachel,

We can first solve for theta by rearrangement and taking the inverse of sin (which is sin^-1(x) or arcsin(x))

3 - 6sin(Θ) = 4 - 8sin(Θ)

8sin(Θ) - 6sin(Θ) = 1

sin(Θ) = 1/2

Θ = sin^-1(1/2)

You correctly identified that one solution to this is π/6, however, the next solution in this set is actually going to be π/6 + 4π/6 (simplified as π/6 + 2π/3 or 5π/6). Notice that this solution lands us in the SECOND quadrant, where the value of the sine of this solution is correctly 1/2. These are the only 2 solutions because sin is negative in the third and fourth quadrants, and remember this problem has a limitation on what theta can be --> The mistake in your solution was that you proposed π/6 + 2π as the next solution. Notice in the beginning of the problem, they limited the values of theta to be

0 ≤ 𝜃 < 2𝜋

and π/6 + 2π is a solution that is outside of this interval

Answer 2

Good news! You clearly know how to solve the equation correctly!

Your answers are actually correct as general solutions to the equation. However, since the domain is limited from 0 ≤ 𝜃 < 2𝜋, adding the "2𝜋n" to your answer is what is making it wrong.

Within the given domain from 0 to 2𝜋, the only solutions would be x = 𝜋/6, 5𝜋/6.

Answer 3

3 - 6 sin θ = 4 - 8 sin θ

2 sin θ = 1

sin θ = 1/2

θ = π/6 + 2kπ

θ = 5π/6 + 2kπ