4-3 Identities and Algebraic Transformations of Expressions

Hey, it's Ellen! Here's what we did on 11/1/06 (All Saints' Day!) for math:

Summary: This lesson's basically about using what we know about trigonometric functions (the relationships we learned yesterday) and using them to either prove identities or to make one equation look like another one. The basic concept behind it is simple, but it can get a little complex somtimes. So here we go...

There's basically seven steps in either transforming an equation or proving an ID (identity):

1. Start by writing out expressions.
2. Determine difference and similarities.
3. Choose a point of attack!!!
4. Sin and cos approach (this step usually makes solving an equation easier, but may be inefficient at some times)
5. Use algebra technics (such as finding a common denominator, multiplying, etc).
6. Look for trigonometric relationships (fractions usually mean its a quotient ID while something squared usually means its a Pythagorean ID).
7. Repeat until the goal is achieved (like shampooing your hair ;D ).

Now that you know basically what to do, here are some examples:

Easy example:

Make sinx cotx look like cosx.

First off, what are the similaries and differences?
There are no same terms on either side of the equation! So we have to figure out a way to get the same terms. Plan of attack: use the quotient ID for cot!
Now it looks like this:




A little more challenging example is:



Now sometimes, you'll be proving ID's. Here's an example of a simple problem where the Pythagorean ID of sinx is proved:



*Note: The book recommends that when solving these types of equations, it's wise to only focus on one side of the equation and solve only that portion until it looks like the other portion. however, Mr. French thinks it's okay to work with both sides of the equation as long as in the end the two are identical. He believes that it still shows that you understand the concept.

**IMPORTANT NOTE: NEVER move parts of an equation from one side to another. NEVER. This violates the point of proving an ID!


Here's a complex example of proving an ID:



Another really complex example is:



It was hard finding a website that deals with this subject directly. But here's one. Click Here

One last thing before I go, here's some pictures of my new baby cousin Andrew!! Isn't he soo adorable?!?!



*yawn*



haha what an interesting expression xD



:)




OH, I ALMOST forgot! LUCY you're up NEXT! G'luck!

Random Note: Did you know tomorrow's National Deviled Egg Day--November 2! lol weird...o.O