Date: Tue, 22 Nov 1994 18:55:35 -0800 (PST) From: Dean Yuan Subject: question I'm a high school senior enrolled in our internet class. One of our projects is to ask you a math question. Your response would be greatly appreciated. What is the definition of sin, and the other trig functions? I understand how to get it, but why was it chosen to be like that? How come all triangles have this property? Does it have something to do with the fact they all have 180 degrees? Your replies are greatly appreciated. Dean Yuan
Date: Mon, 28 Nov 1994 10:59:23 -0500 (EST) From: Dr. Sydney Subject: Re: question and help Dear Dean: Thanks for writing Dr. Math! We are a group of about 15-20 college students and professors at Swarthmore College right outside of Philadelphia. At any given time, we try to have at least one of us logged on to answer questions. We are glad you wrote with your question. Sin is defined to be function that maps an angle to the y coordinate of that angle's intersection with the unit circle. Cos is defined to be the function that maps an angle to to x coordinate of that angle's intersection with the unit circle. When we choose it like this we get lots of nice properties like sin^2(x) + cos ^2 (x) = 1. When you ask "how come all triangles have this property?" I assume you are referring to the fact that for all right triangles (triangles that have one 90 degree angle), sin x = opposite/hypotenuse, cos x = adjacent/hypotenuse, etc.? Well, this is a result of the definition of sin and cos. See if you can figure out why it follows. Here's a hint: draw a circle around the triangle in which the vertex that contatins the angle you are working with is the center of the circle and the circle has a radius equal to the length of the hypotenuse. Please write back if you have any problems with this! Thanks for writing. --Sydney Date: Wed, 30 Nov 1994 11:52:01 +0000 From: Dr. Math Organization: Swarthmore College Math Doctors Subject: Re: question Hey Dean, I'm glad that you are in an internet class and learning how to use the vast resources that it has to offer. Here are a few comments about sin. It can be used in many ways but at it heart I think that sin can be thought of as a ratio of sides of right triangles. I assume that you have seen the definition of the sin of an angle in a right triangle to be the length of the opposite side divided by the length of the hypotenuse. Or sin = opp/hyp. Have you ever studied similar triangles? For any two similar triangles the ratio of their sides is the same. So for any two right triangles with the same angles they have the same sin no matter how big they are. So sin can be thought of as just dependent on the angle. That is why we say sin of 30 degrees even when we don't know what the sides are. I hope that this helps some. There are lots of other ways to think about sin and lots of other ways to use it, but I really think about sin in its most basic form as describing the ratio between the sides of a triangle. For example, do you know the Law of Sines? Check it out and see if you can tell how it relates to this stuff. Ethan Doctor On Call
Search the Dr. Math Library:
Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.