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### Double and Half-Angle Identities

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Date: 12/08/2000 at 08:42:15
From: John Gardner
Subject: Double and Half Angle Identities

Hi Dr. Math:

We are currently learning about double and half-angle identities in my
pre-calculus class. I was just wondering why we are learning them.
What purpose do they serve? Why can't we just take the sine, cosine or
tangent of double or half of the actual angle?

Thanks.
John
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Date: 12/08/2000 at 11:41:37
From: Doctor Rick
Subject: Re: Double and Half Angle Identities

Hi, John.

Here are several reasons, looking at it from different angles (no pun
intended). I'll put them in order from least to most convincing, in my
mind.

1. Historically, identities such as these were used by Ptolemy of
Alexandria to generate the first "trig tables." Your calculator uses a
different means to compute sines and cosines, but it's good to be
aware that those numbers don't just come out of the air; they need to
be calculated, and trig identities can be used to do so.

2. The difference between calculating sin(22.5) and using the trig
identity for sin(45/2) is the difference between an approximation and
an exact value. A calculator always gives you an approximation to the
sine or cosine of an angle, not the exact answer. (Well, sometimes
it's exact - for instance, sin(30 degrees) = 0.5 is exact - but the
calculator can't tell you so.) In pure mathematics we put a high
premium on exactness: an approximation, no matter how good, is an
entirely different beast from an exact solution.

3. You can only use the calculator approach when you have a specific
angle - a number of degrees or radians - to deal with. It is useless
when you are dealing with variables. If you are in the midst of
solving an equation and the equation has sin(x/2) on one side and
sin(x) on the other, you need to use a trig identity to write the one
in terms of the other. Then, letting y = sin(x), you get an equation
in the variable y, which you can solve.

In other words, the trig identities aren't really necessary for simple
calculations, but they show their worth when you use them as tools for
algebraic manipulation.

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

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