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### Trigonometric Identities

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Date: 11/12/97 at 18:16:50
From: Steven Schoenherr
Subject: Trigonometric Identities

I have a huge test coming up on trig identities and was wondering what
the best way to solve verification problems would be.  Also, do I
really have to memorize all of the Identities in order to understand
them?

Thanks,
Steven Schoenherr
```

```
Date: 11/12/97 at 21:00:46
From: Doctor Wilkinson
Subject: Re: Trigonometric Identities

There are some identities that you have to know, and others that you
should be able to derive from the ones you have memorized.

I divide the essential identities into three groups:

(1) Identities that express all the trig functions in terms of the
sine and cosine:

tanx = sinx/cosx
cotx = cosx/sinx
secx = 1/cosx
cscx = 1/sinx

(2) sin^2(x) +cos^2(x) = 1

This is just the Pythagorean Theorem expressed in terms of
trigonometry.

Using (1) and (2) you can prove a whole bunch of other identities, and
this saves you having to memorize them.  For example,

sec^2x - tan^2x =
1/cos^2x - sin^2/cos^2x   (from group 1)
=  (1 - sin^2x)/cos^2x
=  cos^2x/cos^2x         (from 2)
=  1

(3) The addition formulas for sine and cosine:

sin(x + y) = sin(x) cos(y) + cos(x) sin(y)    and

cos(x + y) = cos(x) cos(y) - sin(x) sin(y)

From the formulas in (3) and the formulas in (1) you can derive
addition formulas for the other functions such as the tan. You can
also get the so-called double-angle formulas like

sin(2x) = 2sin(x) cos(x)

I also recommend doing a lot of practice problems, but keep the
memorization down to just the identities I've mentioned already.

-Doctor Wilkinson,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
College Trigonometry
High School Trigonometry

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