The first thing we did in class was go over question 16 from exercise 4 from homework the night before. The question asked where the coordinates were for a) cos (θ + π/2) and b) cos (θ - π). From the previous question, 15, we had figured out that P (θ) was (-3/5, -4/5). So to arrive at answer a) you have to add 90º but Mr. K. explained to us that doing that is the same as (x, y)--R 90º---> (-y, x). So (-3/5, -4/5) would become???...of course! You guessed it! (4/5, -3/5).
For b) you would rotate it 180º. You would have the same number for coordinates but different signs. (x, y)--R 180º-->(-x, -y). (-3/5, -4/5) would become (3/5, 4/5).
Then, we had a quiz on circular functions. There were four questions that we had to do in a given amount of time. The first question was converting degrees into radians. The second question was converting radians into degrees. The third question was stating which quadrant each angle was in. The fourth question was finding what P (Ө) was.
After the quiz was over, we marked it in class and Mr. Kuropatwa taught us how he marked it by telling us how we can gain or lose marks.
One thing Mr. K. stressed was the lovely equal sign. He said not to put equal signs everywhere or leave a string of them lying around. He said there must be something on both sides of the equal sign.
You could also lose marks for doing that!
Aside from that he also distinguished between arithmetic and concept errors. An arithmetic error is when you write something and write it wrong the second time.
For example: You write + 3 in one line and in the next you accidentally write + 30. A concept error is when you do not know what you are doing. An example is trying to solve an equation and you don’t know how to solve it, so you write something but it is wrong.
Something else to consider is NOT TO FORGET writing your units because you could lose marks for that, especially on the exam.
Decimal points. Mr. K. said we have to round to four decimal places unless it is indicated in the question to round to a specific decimal place. Note: for the exam we will be rounding to three decimal places.
So from that quiz, basically to sum it all up we learned how to be a "detective" in marking quizzes/tests/exams.
Things to remember:
-when possible to give exact values: DO IT!
-don’t round to decimal points UNLESS indicated in the question.
-show ALL WORK.
Afternoon:For the afternoon class we started off by solving questions on the board given to us by Mr. K.Solve for x; [0, 2π]:
2cos²x – cosx – 1 = 0
(2cos x + 1) (cosx - 1) = 0
cosx = -1/2 cosx = 1
x=2π/3, 4π/3 x = 0, 2π
We began by factoring the each equation and solving for x.
sec²x + 2secx = 0
secx (secx + 2)=0
secx = 0
secx = -2
cosx = -1/2
x = 2π/3, 4π/3
3)4sin²x = 1
(2sinx - 1) (2sinx + 1)=0
sinx = ½ sinx= -1/2
x = π/6, 5π/6, 7π/6, 11π/6
[remember: this method makes Mr. K. smile =) and you never have to worry about the other roots because it gives you both]
That was fairly simple or so we thought. Mr. K. then asked us what were to happen if we were asked to solve for x with the interval as all the real numbers. So with question 3 Mr. K asked us what the sin of 13π/6 would be and we said it would be ½. He then explained to us that there would be an infinite number of possibilities for the angle of sin=1/2.
So we got:
x = π/6 + 2kπ; kЄI
x = 5π /6 + 2kπ; kЄI
x = 7π /6 + 2kπ; kЄI
x = 11π /6 + 2kπ; kЄI
*Lauressa noticed a pattern to put ± infront of π/6 + 2kπ; kЄI, by doing that that could eliminate writing x = 11π /6 + 2kπ; kЄI. Also, if you put ± infront of 5π /6 + 2kπ; kЄI that could eliminate writing x = 7π /6 + 2kπ; kЄI.
Mr. K said we could think of it as half circles instead of the full circle by adding kπ instead of 2kπ. You would then get x = π/6 + kπ; kЄI and x = 5π /6 + kπ; kЄI.
*C-box [Craig] also noticed a pattern which was ± π/6 + kπ; kЄI, which eliminates writing all four above. Mr. K. said that if you understand how to write it that way then you can do it but if not write it the way he originally taught us.
That then led to the block of wood! DUN! DUN! DUN! [cue scary music!] Just kidding! Okay, so the block of wood Mr. K. showed us had three different rectangular faces. He said that although each is a different representation of the wooden block it is all the same block. That was his analogy for solving problems. That there can be different representations for one concept. Like, sin2x = 1/2. It can be represented graphically with a graph, of course, symbolically with an equation and numerically with a table of values. So to solve sin2x = ½ we let θ = 2x.
We did that then Mr. K. said that since it said 2x there would be two answers for each [however many x there is/are there will be that many multiples, thereof] and we got:
sinθ = ½
θ = π/62
x = π/6
x = π/12
θ = 5π/6
2x = 5π/6
x = 5π/12
θ = 13π/6
2x = 13π/6
x = 13π/12
θ = 17π/6
2x = 17π/6
x = 17π/12
Mr. K. then said all of this would make more sense on Monday because he will show us graphs. After all of that we wrote in our math dictionary for the first time. It was so exciting. Seriously! haha =)
[math dictionary, click link for image] http://img505.imageshack.us/img505/9599/mathdictionarynz4.gif
Okay that's all of this lengthy post folks! Sorry that it was long! =Þ...and remember: don't ever write x = 2x, use colours in your math dictionary, do exercise five, we may be getting a smart board sometime this week and Aichelle rocks!!!...just kidding about the last part..haha. =Þ
Have a good day, do your homework and if I have failed to leave information out of this blog post or if I have made a mistake anywhere, please tell me so I can fix it toute de suite.