Double and Half-Angle IdentitiesDate: 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/ |
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