Tuesday, March 18, 2008

More Pythagorean Identities

K hi guys. My turn to blog for the day or should i say...night.

So we started the morning class off by watching that article 13 video. After that it was straight to math. Today's math class was a workshop since many of us enjoy doing this activity.

Anyways, the lesson for today had a great deal of emphasis on...elegance in showing our work. By elegance, I mean the quickest and most direct way of solving or proofing an identity. The formulas above help a great deal in achieving this elegance. To make things elegant we can change everything into sine, cosine, or any of the other identies to solve faster. Like the question on slide 3. Work did not need to be shown because it is already proven that tan^2a is = sec^2a-1. Showing work is not bad but we are trying to be elegant. Have I overused that word yet? Where was I...oh yeah. Elegance however does not mean skipping steps! Work will need to be shown if the formulas above aren't present in the question. Another thing about the picture above, it is not needed to memorize all those formulas. They are all derived from sin^2x+cos^2x=1. That is the only important one to remember.

This lesson also focused on do's and dont's when solving an identity. The first thing we learned was the "Great Wall of China", which is the line we put under the equal sign. This line can't be passed! This is because it is not an equation. We can't do things like multiplying or adding to balance both sides. However we can "algaebraically massage" the seperate sides into something we can work with. Another don't is putting equal signs, instead we just show the work going down. Putting an equal sign is a no no because we don't know till the end of the work if one side is really equal to the other side. We were also shown some strategies to solving identities, which are above.

Then we were given questions like these, where we could apply those strategies. For the first question, we multiplied top and bottom by 1+sint because we can see 1-sint is a factor of a difference of squares. We multiply it by its other factor to make things easier. Now the top is (1+sint)cos and the bottom (1-sin^2t). The bottom can be changed to a simpler form which is cos^2t! We know that sin^2t + cos^2t=1 without proving it. So after moving sin^2t to the other side we'll get cos^2t=1-sin^2t. That is elegance folks! Then we just continue simplifying it as much as possible. When we're all done we put Q.E.D to indicate we are finished. Q.E.D is some latin word, which I can't remember. So don't ask XD.
The rest of the questions we're mostly the same but using different strategies like factoring and such. Doing these questions need a great deal of practice to get better! Alrightey, there's my scribe post. I hope I included all the important things so yeah. I shall see everyone tmr. Bye bye.
Whoops forgot to name the next scribe. You know who you are! AnhThi. Also, why won't my paragraphs seperate up there...oh well.

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