Verifying Trigonometric IdentitiesDate: 07/12/2002 at 08:10:05 From: Dianne Graham Subject: How do you verify a Trigonometry Identity? How do you verify a Trigonometry Identity? I looked in my textbook and the directions were not clear. Date: 07/12/2002 at 17:24:22 From: Doctor Ian Subject: Re: How do you verify a Trigonometry Identity? Hi Diane, The idea is to manipulate one or both sides of the identity until you have something that is obviously true, e.g., sin(x) = sin(x). It might be instructive to see how you can _create_ an identity. Start with something like sin(x) = sin(x) Now, csc(x) = 1/sin(x), so we can write sin(x) = 1/csc(x) And tan(x) = sin(x)/cos(x), so we can write tan(x)cos(x) = 1/csc(x) And cos^2(x) + sin^2(x) = 1, so we can write sin^2(x) + cos^2(x) tan(x)cos(x) = ------------------- csc(x) sin^2(x) cos^2(x) = -------- + -------- csc(x) csc(x) cos^2(x) = sin^3(x) + -------- csc(x) Now, this isn't a very elegant identity, but it illustrates the point: By starting with a trivially true statement, and exchanging simple expressions for more complicated equivalent expressions, you can build up a statement that must be true, but isn't obviously true. It's very much like what you do to set up an equation to be solved in algebra: http://mathforum.org/library/drmath/view/57265.html Verifying an identity is a matter of going in the opposite direction. Usually you start by converting everything to sines and cosines: cos^2(x) tan(x)cos(x) = sin^3(x) + -------- csc(x) sin(x) cos^2(x) ------cos(x) = sin^3(x) + -------- cos(x) 1/sin(x) And then you use the tricks - e.g., factoring, cancellation - that you were supposed to have learned in algebra: sin(x) = sin^3(x) + sin(x)cos^2(x) = sin(x)(sin^2(x) + cos^2(x)) sin(x) = sin(x) Often the key to verifying an identity is to recognize that something like sin(x)cos(y) + cos(x)sin(y), can be replaced by sin(x+y) There's no magic formula for doing these kinds of problems. It normally involves a lot of trial and error, since you're essentially untying a knot that someone else tied, often just because he thought of a tricky way to tie one. Does this help? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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