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!
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