# Math Help - Finding the x of a sine equation!

1. ## Finding the x of a sine equation!

20 = 8sin((pi/6)(x+2))+16 ........... subtract 16
4 = 8sin(pi/6x+pi/3) ................ divide by 8
1/2 = sin(pi/6x+pi/3) ..............arcsin1/2 to get rid of sin and get pi/6
pi/6 = pi/6x+pi/3 .......... subtract pi/3
-pi/6 = pi/6x ....... divide by pi/6 to get x
-1 = x

My question is: Is there any way to find out the other value of 1/2 sine except looking at the unit circle, which is 5pi/6?

1/2 = sin(pi/6x+pi/3) .. arcsin1/2 to get rid of sin and get 5pi/6
5pi/6 = pi/6x+pi/3 ... subtract pi/3
pi/2 = pi/6x ....... divide by pi/6 to get x
3 = x

And, how would I do it if the y value is 10?

10 = 8sin((pi/6)(x+2))+16

I don't know how to go through the steps to find the one of the values for sine -3/4 (after you go through the steps)
The only one I found was 8.3803207 for x.

By tracing the sine graph on my calculator, I was able to get the answers 5.6196793 and 8.3803207

2. Hello, dayoung!

I would solve it like this . . .

We are given: . $8\sin\left[\frac{\pi}{6}(x+2)\right]+16 \:=\:20$

Subtract 16: . $8\sin\left[\frac{\pi}{6}(x+2)\right] \:=\:4$

Divide by 8: . $\sin\left[\frac{\pi}{6}(x+2)\right] \:=\:\frac{1}{2}$

Take arcsine: . $\frac{\pi}{6}(x+2) \:=\;\begin{Bmatrix}\frac{\pi}{6} + 2\pi n \\ \frac{5\pi}{6} + 2\pi n \end{Bmatrix}\quad\text{ for any integer }n$

Multiply by $\frac{6}{\pi}\!:\;\;x + 2 \;=\;\begin{Bmatrix}1 + 12n \\ 5 + 12n \end{Bmatrix}$

Subtract 2: . $x \;=\;\begin{Bmatrix}\text{-}1 + 12n \\ 3 + 12n \end{Bmatrix}$

3. How would I do it if the y value is 10?

10 = 8sin((pi/6)(x+2))+16

I don't know how to go through the steps to find the one of the values for sine -3/4 (after you go through the steps)
The only one I found was 8.3803207 for x.

By tracing the sine graph on my calculator, I was able to get the answers 5.6196793 and 8.3803207