Double and Half-Angle Identities

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

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/