Thursday, September 13, 2007

Pythagorean Identities

Starting with the fundamental trig identities that I've already covered, a little bit of derivation leads to several new identities that will always hold true, and are especially useful in simplifying more elaborate equations.  Expanding upon my last post, there are a few other identities that can be derived from the Pythagorean Identity, which also go by the same name to describe the group of them, due to how they are deduced by making use of the his famous Theorem.  (Click here to get a great explanation about these Pythagorean Identities.)

If we start with the first Pythagorean identity, which I already explained:

Sin2(ɵ) + Cos2(ɵ) = 1

we can divide each side by the sine term to give something new.

[Sin2(ɵ) + Cos2(ɵ)  ] / Sin2(ɵ)  = 1 / Sin2(ɵ)

The first term (sin / sin) reduces to 1:

1 + Cos2(ɵ)  / Sin2(ɵ)  = 1 / Sin2(ɵ)

The remaining terms can be now be simplified.  We can apply the laws of exponents to group the terms and express them in a slightly different way, which allows us to apply additional trig identity substitutions to express things in terms of inverse functions:

1 + [Cos(ɵ)  / Sin(ɵ)]2  = [1 / Sin(ɵ)]2

1 + Cot2(ɵ)  = Csc2(ɵ)

This is the second of the three Pythagorean identities.  Hopefully my explanation is clear enough to understand, and you can follow along with my derivation of it.  As I always say on this site, if you understand the derivation, then you don't have to worry about making mistakes in memorizing the actual formula.  This is a perfect example of that, because the starting point is very easy to remember, but even I sometimes will confuse myself if I try to write down these additional identities without thinking about how to get there.  Learning the derivations of identities, formulas, and equations will always help you!

As I said, there are three Pythagorean identities.  This third one is derived in a similar manner to what we have already done, using the same sort of logical steps but dividing instead by the cosine term (rather than the sine term).  I am not going to write down all of the steps here, but I highly recommend that you grab a pencil and paper and quickly work through it, following along with the process that I outlined above.  There are only a few steps to work through, and in the end, this is what you should end up with:

1 + Tan2(ɵ)  = Sec2(ɵ)

As it is with many identities and equations in mathematics, it doesn't take very many steps to make something look completely different, as I've shown here.  I hope that you've been able to follow along and that everything makes sense.  If the steps in the derivations confused you at all, then I suggest going back to my posts on the basic trig functions, and working with those until you are comfortable.  The steps I have applied in these transformations were very simple substitutions, and I think that the most confusing part knowing what to look for in the first place.  With practice, you will learn to see patterns and recognize where you can apply various rules to make things simpler.

There are still several other trigonometric identities which I will show you how to derive in coming posts, many of which require the same types of thinking.  Stay tuned for more tutorials that explain things like the sine law and cosine law!

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