solve the equation for exact solutions over the interval (0 degrees,360 degrees)

Question

3sinθ=sinθ-√3

Answer 1

In order to solve 3sin(x) = sin(x) - sqrt(3), we can start by treating sin(x) like it is just a single variable, because the first half of this problem is really only algebra. So lets get the sin(x) terms onto one side by subtracting sin(x) from both sides:

3sin(x) - sin(x) = -sqrt(3)

If you notice, 3sin(x) and sin(x) are like terms since they both have sin(x) in them, so combining them gives us:

2sin(x) = -sqrt(3)

Now we can divide by 2 on both sides to get sin(x) by itself:

sin(x) = -sqrt(3)/2

Now that we have the value of sin(x), we can go and find the angles. We know that sin(60) = sqrt(3)/2, so our reference angle is 60 degrees. Since our value came out negative, x has to be in Quadrant III or Quadrant IV, because those are the two places where sine is negative.

In Quadrant III we take 180 + 60, which gives us 240 degrees, and in Quadrant IV we take 360 - 60, which gives us 300 degrees.

If you plug either one back into the original equation, both sides come out to -3sqrt(3)/2, so they both check out. There are no other answers in the interval because sine only reaches -sqrt(3)/2 twice in one full turn.

So, our final answer is x = 240 degrees and x = 300 degrees.

Answer 2

The stated problem looks like 3sin(a) = sin(a) - 3^(1/2) if so, then 2sin(a) = -3^(1/2) = -1.732

or sin(a) = -.866 a = sin-1(-.866) a = -60 degrees= 300 degrees

but in the 3rd quadrant a = 240 degrees

sin(300) = sin(240) = -.866