Trig Functions - "When wil object be 9cm below 0?"

[TN17]

1. The problem statement, all variables and given/known data
Here is the background information:

A weight hanging from a spring is set in motion by an upward push. It takes 10 s for the weight to complete one cycle from moving 12 cm above 0, then dropping 12 cm below 0, then returning to 0.

Here is the question:

d) In the first 10 s, when will the height of the weight be 9 cm below 0?


2. Relevant equations
I found the equation to be y=12sin (Pi/5)(x)
Since k=2Pi/10 = Pi/5 and the vertical stretch is 12.


3. The attempt at a solution
I set y=-9 because the weight is below 0, and solved for x, but I didn't know how to continue from there.
-9 = 12sin(Pi/5)(x)

There are 2 answers, 6.3 s and 8.7 s

[HallsofIvy]

1. The problem statement, all variables and given/known data
Here is the background information:

A weight hanging from a spring is set in motion by an upward push. It takes 10 s for the weight to complete one cycle from moving 12 cm above 0, then dropping 12 cm below 0, then returning to 0.

Here is the question:

d) In the first 10 s, when will the height of the weight be 9 cm below 0?


2. Relevant equations
I found the equation to be y=12sin (Pi/5)(x)
Since k=2Pi/10 = Pi/5 and the vertical stretch is 12.


3. The attempt at a solution
I set y=-9 because the weight is below 0, and solved for x, but I didn't know how to continue from there.
-9 = 12sin(Pi/5)(x)
Well, solve the equation for x by the usual way- "unpeel" what has been done to x.
First, divide both sides by 12:
\frac{-9}{12}= -\frac{3}{4}= sin(\pi x/5)
Now, use the inverse function, arcsine, to get rid of the sine function:
\frac{\pi x}{5}= arcsin(-3/4)= -0.8481
is the "principal solution" given by a calculator. Of course, we want x to be positive so we use the fact that sin(\pi- \theta)= sin(\theta) and, of course, sin(2\pi+ \theta)= sin(\theta). \pi- (-.8481)= 3.1416+ .8481= 3.9896 and 2\pi+ (-.8481)= 6.2832- .8481= 5.4351
From
\frac{\pi}{5}x= 3.9896
and
\frac{\pi}{5}x= 5.4351
we get
x= (3.9896)\left(\frac{5}{\pi}\right)= 6.3496
x= (5.4251)\left(\frac{5}{\pi}\right)= 8.6502
which round to the values you give.
There are 2 answers, 6.3 s and 8.7 s