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### Sine, Cosine, Tangent

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Date: 03/13/2001 at 00:49:35
From: kelsie gale
Subject: Sine, cosine, and tangent

What are sine, cosine, and tangent?
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```
Date: 03/13/2001 at 11:25:42
From: Doctor Rob
Subject: Re: Sine, cosine, and tangent

Thanks for writing to Ask Dr. Math, Kelsie!

Sine, cosine, and tangent are functions that take an angle and give
you a corresponding real number.

Let A be an acute angle, and consider the following diagram:

B"_,-'
B'_,o'
B _,o'  |
_,o'  |   |
_,-'  |   |   |
_,-'      |   |   |
_,-'          |   |   |
_,-'              |   |   |
A o-------------------o---o---o-----
C   C'  C"

Here <ACB, <AC'B', and <AC"B" are all right angles.

The three triangles ABC, AB'C', and AB"C", are similar. (Why?) That
means their corresponding sides are in proportion to each other. This
tells you that, for example,

CB/AB = C'B'/AB' = C"B"/AB"

That means that in a right triangle, no matter its size, the ratio of
the leg opposite the angle A, to the hypotenuse, is a certain
constant.  That ratio is called the sine of the angle A, and is
written sin(A).

Similarly, the ratio of the leg adjacent to the angle, to the
hypotenuse,

AC/AB = AC'/AB' = AC"/AB"

is a constant, called the cosine of the angle A, and written cos(A).

Similarly, the ratio of the leg opposite to the angle, to the leg

CB/AC = C'B'/AC' = C"B"/AC"

is a constant, called the tangent of the angle A, and written tan(A).

These three functions can also be defined for zero, right, obtuse,
and other angles using the diagram at the top of the following Web
page from our Frequently Asked Questions (FAQ):

Trigonometry Formulas
http://mathforum.org/dr.math/faq/formulas/faq.trig.html

- Doctor Rob, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Trigonometry

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