SOLUTION: Find all values of x over -2pi and under 2pi. The problem is sin x = -.7 I know that you first use inverse sine to find the first angle but I don't know how to find the other an

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Question 412125: Find all values of x over -2pi and under 2pi. The problem is sin x = -.7
I know that you first use inverse sine to find the first angle but I don't know how to find the other angle measures. Please help! Thank you!(:
Answer by jsmallt9(3758) (Show Source):
You can put this solution on YOUR website!
First we will find the general solution (i.e. all the angles whose sin is -0.7). Then we will use that to find the angles between and .

To find the general solution we first need the reference angle. And for this we need to use our calculator and the inverse sin function.
Make sure your calculator is set to radian mode. (If you don't know how to do this, then see "Calculator mode" at the end.)
Find sin^-1(0.7) Note that it is positive 0.7 not negative! (We will address the negative of -0.7 shortly.)

You should get a reference angle of 0.7753974966107531. (Feel free to round this off to whatever accuracy you and/or your teacher think is acceptable.) If you did not get a number close to 0.7753974966107531, then either your calculator is not in radian mode (See "Calculator mode" at the end.) or you entered the inverse sin incorrectly (perhaps you entered the minus sign?).

Now we will address the issue of the minus of -0.7. The sin function has negative values for angles that terminate in the 3rd and 4th quadrants. SO we are looking for angles in the 3rd and 4th quadrants whose reference angle is 0.7753974966107531.

In the 3rd quadrant, we add the reference angle and :
0.7753974966107531 + which is approximately 3.9169901502005462
In the 4th quarter, we subtract the reference angle from :
- 0.7753974966107531 which is approximately 5.5077878105688332

There are an infinite number of angles whose sin is -0.7. But they all have a reference angle of 0.7753974966107531 and they all terminate in either the 3rd or 4th quadrants. To express these infinite solutions (without listing each one) is done is a special way:
x = 3.9169901502005462 + *n
or
x = 5.5077878105688332 + *n
The first equation handles all the angles in the 3rd quadrant and the second equation handles all the angles in the 4th quadrant. In these equations, the "n" can be any integer. With these infinite possible values for "n" we are able to express the infinite solutions for the angles whose sin is -0.7.

You are only interested in the angles between and . So we just have to figure the right integers for "n" that give us an "x" that is in that range. For your range use n = 0 and n = -1 (in each equation). This should give you 4 values for x, 2 positive and 2 negative. I'll leave it up to you to calculate these values.