Tangent! It had something to do with tangent.... O right! There were several purple slides back to back and we taught ourselves all about double angel identities.

We first proved a double angle tangent identity.

Now don't fear if you look at this and fear for your life, it's really not that difficult! We start off with this:

To get to the next line as we all know tangent is equal to sine over cosine so we simply replace tangent and using our knowledge of sin and cosine double angels we expand it all out to:

Now we're trying to get it in forms of tangents. So you look and see that you have some sines and think tangent is sine over cosine so maybe if i divide EVERYTHING by cosine something wonderful may happen. So your multiplying (because we don't divide) by 1 over cosine or secant!! This will result in:

So you all see how you get that?? You divide each of those by cosine alpha cosine beta. In the first one, both cosine betas reduce to one leaving you with tangent alpha. The second, the cosine alphas reduce to one leaving you with tangent beta. On the bottom, the cosine alpha cosine betas reduce to one. Leaving only the sines to be divided by cosine alpha cosine beta leaving you with tangent alpha tangent beta and zip bing bang.... QED!!!!

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To prove tangent alpha MINUS beta we use the clever idea of instead adding the number we subtract a negative number. This is in fact the same thing!!!!! So where ever there was tangent beta in this proof we simply switch it for a negative beta resulting in:

Then finally ending up with:

WARNING!! We can only do this because we have first proved the identity of tangent alpha plus beta!!!!!

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The last thing we taught ourselves was the double angel identities of sin(2 theta) and cos(2 theta). Now the clever idea here is to see that 2 theta is the same as theta plus theta. We then use our previous knowledge of the double angle identities for sine and cosine to do the following:

Sine is straight forward and you get only one different identity. For cosine on the other hand there are 3 different ways to manipulate it. Once you get it into the form cosine squared minus sine squared, you can use a Pythagorean identity and substitute in either 1 minus sine squared or one minus cosine squared for cosine or sine respectively. If your unsure of which you need to use have no fear because no one knows which you will need! The only way you will know is by looking at the context of your question!

Well that was the class and I'm signing off i think we already know that P.C. (pokemon champion) is the scribe for todays class. so have fun everyone, happy blogging and don;t forget pre test tomorrow and test Thursday! Study up, do your BOB and find your delicious websites!!! May the force be with you all!

THANK YOU!THANK YOU!THANK YOU!THANK YOU!THANK YOU!THANK YOU!THANK YOU!THANK YOU!THANK YOU! I can't thank you enough.=P

ReplyDeleteLOL! great post BTW.

lmfao sall good pj XD not nearly as good as yours haha

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