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### Trig functions

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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?

Dean Yuan
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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

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
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Associated Topics:
High School Trigonometry

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