Sines and equations

Date: Sat, 3 Dec 1994 23:28:43 -0500
From: Anonymous
Subject: Sines and equations...

Hi -

I am a tenth grade student taking AMF (advanced mathematical functions)
and was given the problem:

0<=X<=150
When does the Sin 8X = Sin X?

I had no problem doing this on my graphing calculator, and finding the
intersects between the functions, but is there any way to do this by using
some sort of equation?

Thanks in advance,
Steve Toub

Date: 5 Dec 1994 17:50:22 GMT 
From: Dr. Math
Organization: Swarthmore College
Subject: Re: Sines and equations...

Hello there Steve!

Let me tell you right away that we here at Math Headquarters LOVED 
THIS PROBLEM!! 

The key here is to think about the definition of the Sine function as it
relates to the unit circle.  If you've got a ray coming out from the origin
that makes an angle of r degrees with the x-axis, the Sine of r is equal
to the y coordinate of the point where the ray intersects the unit
circle.  You'll have to draw a picture in order to really see what's going
on.  Have you seen the unit circle before?  If you haven't, ask your
teacher about it, or write back to us about it.  

Draw the unit circle on the x-and y-coordinate axes, and make a ray going
out from the origin.  Label the angle that this ray makes with the x-axis
with the letter r (I would have used x, as in your equation, but I thought
that might get confusing, with this x-axis floating around).

So what we're looking for is an angle r such that when you multiply it by
8 and subtract some multiple of 360 degrees, you get the same angle
again.  Does that make sense to you?  There's another way to find some
more solutions, though.  Notice in your picture that the Sine of r is
equal to the Sine of 180-r, since they both have the same y coordinate. 
So we can also look for angles r such that 180 - r is equal to 8r minus
some multiple of 360.

So we can get the following equations:

r = 8r
r = 8r - 360
r = 8r - 720
r = 8r - 1080

180 - r = 8r
180 - r = 8r - 360
180 - r = 8r - 720
180 - r = 8r - 1080
180 - r = 8r - 1440

And if you solve these equations, you should get the answers you're
looking for.  And here's another thing: have you seen radians (another way
to measure angles) before, as opposed to degrees?  Equations generally get
simpler, with smaller numbers, when you use radians.

Anyway, I hope this helps you!  Write back if you need anything else!