April 26, 2010 - the greatest day in the history of mankind

Happy birthday to Annieeee, happy birthday to me!

For a warmup today we had about 25 minutes to...

1. Graph . We wanted to be able to do this problem on a non-calculator section. This problem required the use of the double angle identity for tangent. The graph was basically that function compressed by a factor of 2.

2. Solve the equation on the interval . We also tried to be able to do this one without a calculator. This one required the use of the double angle identity for sine. Then you can factor out the cosx and solve for the cosx and sinx values - then find the spots on the unit circle where those values are true.

3. Find the exact value of cos150˚.
We first thought of doing this by splitting up the cos150 into parts for a sum identity- 60˚ and 45˚. Maci pointed out that you could also transfer it into radians if that helps you. Mr. O'Brien reminded us that we can also use the difference identity for cosine or a half angle identity. We decided to try the half angle identity. So we split cos150˚ into and then used the half angle identity to progress through the problem. Our answer comes out to be . This is an exact value. Then we compared this answer with the answer the sum identity people got, which was . Turns out, the two values are the same! Mr. O'B challenged us to show why these two values are the same thing - anyone can try this for an IOU.

4. Finally, Mr. O'Brien gave us a simple graph with a line coming from the origin that went through the point (1,2). The angle created by this line was called u. We had to use this graph to find the exact value of cos2u. Again, this one required the use of the double angle identity for cosine. We found the answer to be or . We then tried to do it with technology using arcsin and arccos, which confirmed our answer. Easy peasy pumpkin squeezy.

Today we discussed what's coming up. All we have to do for the next couple of weeks is keep up with the homework - we have homework quizzes on Friday and next Thursday. The big unit test is two weeks from Wednesday. So make sure to keep up with homework, and remember to only use the answers from the book as a checker - if you just copy them down, you won't learn the processes.

We also talked about the homework quiz from last class. Mr. O'Brien handed them back... There seemed to be a general consensus that it was difficult and that we wished we had had more time. The best way to be prepared for these is to just be thorough with your homework and make sure you ask questions and check your answers. Mr. O'B reminded us to go back and check all our quiz questions that we got wrong so we can learn from them. All that good stuff.

Notes from the quiz:
1. Remember to sub back in answers and include the +2πk.
2. Remember to always take the plus and minus of a square root. You don't have to list all four solutions. We could have just used a calculator and then figured out what the answers are, but that can take more time.
4. If you get stuck on the algebra, just solve it with technology and move on. Then come back later. But this one required the use of tangent and secant identities. 0 was an extraneous solution.
5. Using the quadratic equation resulted in . Then you had to remember that u really meant sinx, and you could solve using arcsin. Then you had to subtract both those from π to find the other two answers. This one was tricky tricky.
6... was quite fun. Mr. O'B suggested making a table with x values of 1, 2, 3... 12. Wherever s is larger than 100, those numbers represent the months of January, November and December.
7. Remember when you have a denominator of 12, you can break it up into 3π/12 and 4π/12. Then you can solve using a difference identity.

Next we went over our homework questions from the p.420 assignment - specific questions were #41, #43, #77, and the tan part of #53. We went over each question individually and worked them out together. If you were absent today and need help on those, you can ask someone for their notes.

41.

43.

53.

77. Prove:

FINALLY... we went back to "The Geometry Days", as Mr. O'B fondly refers to them. The next 2 days are going to be spent on the Sine Rule and Cosine Rule. We learned how to derive the Sine Rule in class today. If you need notes, I'd borrow them from Lange. She does pretty colors and everything. Or Nick. He goes everywhere she goes. For the homework tonight you'll need to use the Sine Rule plus the formula for the area of a triangle (half base times height).

Next scribe is Sir Nick!