Double Post?!?!

Hey folks! Ok so this will be the last unit-type post before the test on Wednesday (I think?) and it will be a DOUBLE POST!!!! I'll at least post the important non-math lesson related things here right now as I slowly work on the actual material stuff.

First off, we are now on CYCLE 2 of the scribe list. All the names have been "uncrossed" off the list.

Second, here is the schedule for when we get back to school.

MONDAY, APRIL 6 - WORKSHOP DAY (TO REVIEW AND SUCH)
TUESDAY, APRIL 7 - PRE-TEST FOR IDENTITIES
WEDNESDAY, APRIL 8 - ACTUAL TEST FOR IDENTITIES (Don't forget to do your BOB's and look for "delicious links"!)

LOL! I'm pretty sure that's the schedule but I'm not too sure.

The scribe post is currently done:

-a review of the last two days before Spring Break (done. Day 1:scroll down Day 2: click here!)
-links that can help you study/review for this unit (done)
-extra reminders and such (done)
-a whole mess of videos, most likely (done:4/4)

Lastly, if there's anyone out there that wants to help me with this monstrous task, please comment. Don't worry, if you help, this will count towards your scribe posts and you will be crossed off the scribe list for Cycle 2. Thanks to Dion!

Hopefully this will be the last double scribe post. I hope that we can keep this blog up to date and that we continue to make this blog a rich textbook for people visiting our blog!

~jayp~


P.S. The next scribe is the ~Pokemon Champion~! LOL it's only because I was playing Pokemon earlier. xD Have Fun!
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Okay so here is the actual post.

Hello guys!

I hope you guys enjoyed this "wonderful" Spring Break! xD

The first thing that happened on this Thursday class was a quiz on what we had previously learned on Trigonometric Identities. In case you missed it, or you want to refresh your memory, here is the quiz!


Youtube Version


And here are the answers:


Youtube Version

Haah ok. Now the hard part. For the rest of the class, we discussed where the sum and difference identities came from.

If you can recall, the identities are:

sin(a+b) = sin(a)cos(b) + cos(a)sin(b)
sin(a -b) = sin(a)cos(b) - cos(a)sin(b)
cos(a+b) = cos(a)cos(b) - sin(a)sin(b)
cos(a-b) = cos(a)cos(b) + sin(a)sin(b)

First, we'll start off by proving cos(a-b)=cos(a)cos(b)+sin(a)sin(b).

Recall from the previous class when we figured out that:

*In my written explanations, I will be using "a" to represent "alpha" and "b" to represent "beta".

Just to refresh your memories, when we were finding the coordinates of P and Q, we remembered that the "x" value is the same as the cosine value and the "y" value is the same as the sine value.
Therefore:
P (cos b, sin b)
Q (cos a, sin a)

When we were finding the coordinates of R, we remembered that when you rotate a point (x,y) 90^o counterclockwise, the new coordinates are (-y,x).

Therefore, if point R is point P rotated 90^o counterclockwise, that means that point R must be (-sin b, cos b).

After we "rigidly rotated" (rotate, but didn't change the actual shape) angle ROP so that P lies on the x-axis, we found out the coordinates of R¹, P¹ and Q¹.

Since P¹ lies on the x-axis, this means that the coordinates of P¹ are (1,0).

Since R¹ lies on the y-axis, this means that the coordinates of R¹ are (0,1).

Doing what we did earlier to find the coordinates of P and Q, the coordinates of Q¹ are (cos(a-b), sin(a-b)).

Now, since we rigidly rotated angle ROP, this means that the distance between P and Q and the distance between P¹ and Q¹ are equal. Because, the distances are equal, we can plug the "x" values and the "y" values into the distance formula and say that:
distance PQ=distance P¹Q¹

Here is a video to show the "grunt work".


Youtube Version

Now let's prove sin(a-b)=sin(a)cos(b) - cos(a)sin(b).

To do this, we use the same method that we used for cos(a-b), but instead of using the points P, Q, P¹ and Q¹, we will use the points R, Q, R¹ and Q¹.

Here is the "grunt work" for proving the sine difference identity.


Youtube Version

Now, to prove the sum identities. Both sum identities can be proven using the same clever idea. This clever idea is simply seeing that a+b=a-(-b). It also involves using what we know about even and odd functions.

For the sine sum identity, if we replace sin(a+b) with sin(a-(-b)), this means that whenever we see a "b" in the difference equation, we replace it with a "-b". This would change the equation into:
sin(a-(-b))=sin(a)cos(-b) - cos(a)sin(-b)

Now if you recall that sine and tangent are odd functions and that cosine is an even function, you will also recall that:
sin(-x)=-sin(x)
tan(-x)=-tan(x)
cos(x)=cos(-x)

Using that knowledge, we can even further rewrite this equation as:
sin(a+b)=sin(a)cos(b) - cos(a)(-sin(b))

which simplifies to:
sin(a+b)=sin(a)cos(b) + cos(a)sin(b)
* notice the sign change

For the cosine sum identity, we do the exact same thing.
cos(a-b)=cos(a)cos(b) + sin(a)sin(b)

becomes

cos(a-(-b))=cos(a)cos(-b) + sin(a)sin(-b)

Then by using our knowledge of even and odd functions, that equation turns into:
cos(a+b)=cos(a)cos(b) + sin(a)(-sin(b))

which will simplify to:
cos(a+b)=cos(a)cos(b) - sin(a)sin(b)

Tada! We have just proven the four sum and difference identities!
This is basically what we did this whole class. Knowing some of this may be handy in the future and it's always important for us to understand where the formulas we use in math originate from. I think that this was the whole point of this exercise. Remember, if you ever forget these sum/difference identites.................DO THE SINE DANCE!!!!!!!! LOL. It's the dance craze that's sweeping the......classroom? Haah.

Also, here are some links to help you refresh your memory about this unit or to help you study for the test:
* You need a .pdf reader to view these links.
Identities 1:Product/Quotient
Identities 2: Sum/Difference
Identities Practice Test (a few questions do not deal with Identities)
Identities Practice Test Answers

Well anyways, I'd just like to say a few words before I sign off. I'm terribly sorry for this really late post. I know I had a full week to work on this but I guess my other Spring Break plans just got in the way. Also, I had to do a lot of planning for an English project that is worth 35% of our course mark (without the exam). Please forgive me, I will do my best to have the two videos I'm missing in this post up and running tomorrow.

Peace out, see you guys tomorrow and always do your best!
~jayp~