Date: 05/24/99 at 22:51:25 From: Christy Miller Subject: Trigonometric Identities I just started in Trigonometry with Identities. (I will use @ to mean theta.) I understand how to verify or prove basic identities true such as: sec@ = 1/cos@, but I do not understand how to simplify problems such as sin^2(@) - 1 One example the book gives is that sec^2(@) - 1 simplifies to tan^2(@). I do not see at all where they can get that answer. One other question I had was; where does theta come from? Does it always represent the angle on the x/y axis of a Cartesian plane? Thank you so much for your help.
Date: 05/25/99 at 11:05:17 From: Doctor Rick Subject: Re: Trigonometric Identities Hi, Christy, welcome to Ask Dr. Math. The first example you give of an identity comes straight from the definitions of secant and cosine in terms of the unit circle. This kind of identity is quite important, but another kind comes from perhaps the most widely used theorem in math - the Pythagorean theorem. Take a look at the unit circle: ******* * O * /| * * / | * * 1 / | * * / |sin@ * * / | * * / @ | * *---------------+------+--------* * cos@ * * * * * * * * * * * * * ******* You can see from the right triangle that the Pythagorean theorem, in trigonometric terms, is sin^2(@) + cos^2(@) = 1 This fundamental trigonometric identity is VERY useful in proving other identities. Watch how it is used to simplify sec^2(@)-1: 1 sec^2(@) - 1 = -------- - 1 cos^2(@) 1 cos^2(@) = -------- - -------- cos^2(@) cos^2(@) 1 - cos^2(@) = --------------- cos^2(@) sin^2(@) (Pythagoras!) = -------- cos^2(@) = tan^2(@) >One other question I had was; where does Theta come from? Does it >always represent the angle on the x/y axis of a Cartesian plane? You can always think of the argument of a trig function (theta) as the angle in the unit circle, whenever you need to go back to definitions to understand a trig concept. But theta does not have to be the angle with the x axis, or even any angle that you can measure. The trig concepts that germinate in the unit circle can be transplanted to all sorts of environments. I work with digital (high definition) television. Part of the process of encoding pictures digitally for transmission to your home is something called the Discrete Cosine Transformation. This cosine doesn't measure any angle you can see in the picture; rather, it has to do with how the brightness of the picture varies as you move across the picture. You'd be amazed at the places that trigonometry shows up! - Doctor Rick, The Math Forum http://mathforum.org/dr.math/
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