Thursday Febuary, 25, 2010... Quiz day.

Unfortunately Mr. O'Brien was not here today. But we pushed on without him, learning about the graphs of the functions we had been studying up until now.

We started the class off with a quiz, which I personally thought was more challenging than the previous homework quizs. The problems that I had the most trouble with were 3. and 5. on the first page. I couldn't quite remember the whole of either concept, and without the full understanding of this concept, I was not able to correctly do the problem.

Fortunately, I still have enough time until the test to learn these concepts so that I will not make the same mistake twice.

After the quiz, Mr. O'Brien informed us through the blog that our work for the rest of the class was to a. get into small groups and go over the homework and b. to "Using Geogebra/Grapher/graphing calculator, sketch the graphs of the tangent, cotangent, secant, and cosecant graphs in your notebook (use graph paper and appropriate scales!)."
and then...

"Look at your your six trig function graphs, and determine the period, domain, range, and whether each is even/odd/neither."

On the homework, I had little trouble with the graphs, as for the most part, they could be done using the calculator. However common problems were found on the last of the homework, and that could be beneficial to go over as a class when Mr. O'Brien is back.

After going over the homework, Robin and I graphed the remaining four functions. They looked like this:
Tangent:


Domain: All real numbers except where x is a multiple of π/2 or if it is 0
Range:All real numbers
Cotangent:

Domain: All real numbers except where x is a multiple of π or 0
Range:All real numbers
Secant:

Domain: All real numbers except where x is a multiple of π/2 or 0
Range:All real numbers

Cosecant:

Domain: All real numbers except where x is a multiple of π or 0
Range:All real numbers

(Images courtesy of http://www.intmath.com/Trigonometric-graphs/4_Graphs-tangent-cotangent-secant-cosecant.php)

I think that the domain and the range MAY be correct, but i'm not 100% sure.
The class had a bit of a hard time interpreting the period and even/odd/neither. When you get back we'll probably need help with this. But it was interesting to see what the different graphs look like in comparison to sin/cos.
I personally forget how to determine even/odd/neither completely , and will need some reminders on this.

By the time we had finished domain and range, it was the end of class, and I had no time to review even, odd or neither and noone around me seemed to have a clear grasp or understanding of a period, so I could not work on that either. These would be good subjects to review in class on monday.

HW: The ical doesn't have the hw posted as of now, I'm sure that Mr. O'Brien will correct this when he sees this.

Class Thursday, February 25th

Hi guys-

Sorry I can't be in today. After the homework quiz, please get into small groups to go over the homework due today (the transformation applet is a good one to play with some more if you had trouble graphing with transformations).

Using Geogebra/Grapher/graphing calculator, sketch the graphs of the tangent, cotangent, secant, and cosecant graphs in your notebook (use graph paper and appropriate scales!). What connections can you find between these graphs and the graphs of the sine/cosine waves?

Look at your your six trig function graphs, and determine the period, domain, range, and whether each is even/odd/neither.

When you finish this work, you may begin the homework listed in iCal.

February 23 - Scribe Post

Our warmup today involved taking a quizlet quiz on trig functions and the unit circle. We were allowed to use calculators and/or units circles, but BEWARE, because that opportunity will not be given on quiz day (Thursday!!)

We then moved into HW questions. There were a few questions from page 318 that were tricky...we worked them out in class. This homework should have been done Sunday, Monday, and Tuesday night, since vacation was supposed to be mathematics-free.

A few people were confused about 81 and 83. These involved finding an angle in radians and degrees given its sin, cos, etc. Ill give the example of 81:a...

81) a)
First, remember that sin represents the Y coordinate. Then, find the two places on the unit circle where the Y coordinate is positive one half. These happen to be 30 and 150 degrees.
Therefore, = 30 degrees, 150 degrees, radians, or radians.

Our main topic for the day was Trig Graphing. We started out with a demo on the Ferris Wheel site (linked in previous post). This shows how the relationship between and sine of on a coordinate plane creates a sinusoid wave. This is also known as the sine function.

There are two key geometric features of a sinusoid wave:
-Amplitude: the distance from the middle to the top or bottom of the wave.
-Period: the length of a cycle of the graph (one time around the unit circle).

Example:
, where the amplitude is the absolute value of , and the period is

The variation of these two factors transforms the graph. This is illustrated here: Transformations of the sinusoid wave

HW for today: Check out iCal, and don't forget to study your 198 trig values for the Q-U-I-Z next class.

The next scribe shall be Señor Murphy

Links for Tuesday, Feb. 23

Warm-up quiz
Ferris wheel applet
Transformations of the sinusoid wave

homework?

Hi everyone, I'm a little confused about the homework because of the mixup from last time. It says on iCal that we have (from last class): page 309/1-15 odd, 27-51 odd. Then (for next class): page 309/17-25 odd, 53-69 odd. Does this mean that we basically have to do 1-69 odd? If so, is it all due Friday before vacation?

Links for Wednesday, Feb. 10

Quizlet.com for 98 flash cards of the common trig values

Quizlet.com for all 198 common trig values!

sine function of any angle applet

cosine function of any angle applet

our unit circle applet

Quarter 3 Project

February 8, 2010 MATH FUN

A link that I found helpful and summed up the identities that I also listed below:http://www.sosmath.com/trig/Trig5/trig5/trig5.html

Today we started out with a warm-up which tested our knowledge of the basic trig functions. If you would like to practice this at home it is on the blog.

Next we chatted about patterns:
Sin
0 0˚ 0 Think of them as square roots, and remember: COS is the same exact backwards...
π/6 30˚ 1/2
π/4 45˚ √2/2
π/3 60˚ √3/2
π/2 90˚ 1---- think of it as √4/2

ALSO: tan π/6=√3/3 tanπ/4=1 tanπ/3=√3

1/√3 and √3/3 are the same thing just rationalized

Remember:
Quad I- sin θ + Quad I-A
cos θ +
tan θ +
Quad 2- sin θ + Quad II-S
Quad 3- tan θ + Quad III-T
Quad 4- cos θ + Quad IV-C
A fun way to remember this is All Students Take Calculus

The reference angle- is an acute angle formed by the terminal side x-axis

To solve things such as cot(150˚), look to the reference angle, which is 30˚. Think of the quadrant it is in and you get cot150˚=-cot30˚=-1/tan30˚=-1/1/√3= -√3

Another good tip other than funny form of one is: -3/√3=-√9/√3=-√9/3=-√3

Identities:
Reciprocal: sinθ=1/cscθ cscθ=1/sinθ
cos θ=1/secθ secθ=1/cosθ
tanθ=1/cotθ cotθ=1/tanθ

Quotient: tanθ=sinθ/cosθ
cotθ= cosθ/sinθ

Co-Function: sinθ=cos((π/2)-θ)
cosθ=sin((π/2)-θ)
secθ=csc((π/2)-θ)
cscθ=sec((π/2)-θ)
tanθ=cot((π/2)-θ)
cotθ=tan((π/2)-θ)

Pythagorean: sin²θ+cos²θ=1 then divide both sides by a sin²θ
which makes it: 1+cot²θ=csc²θ
and also: 1+tan²θ=sec²θ

After taking some vigorous notes we then went over the quiz from last time.

HW: Look on the ical and also we have a QUIZ next class

On my mind: I need to memorize. I feel like there are a lot of patterns but I just need to find time to put them all together and really make sense of the patterns.

Links for Monday, Feb. 8

Quizlet.com for 39 trig basics flashcards
Quizlet.com for 98 flash cards of the common trig values
Quizlet.com for all 198 common trig values!
sine function of any angle applet
cosine function of any angle applet
our unit circle applet

Feb. 2, 2010- Scribe Post

Today, we started class by going over Francios and essentially what we covered last class for the people who were at District 3 on Friday. We covered arch length proportions and degrees/radians.






Then, we looked over the radian to degree ratio using the interactive circle, which you can find here: http://www.dudefree.com/Student_Tools/materials/precalc/unit-circle.php
We then raced to see who could get the best time on the scatter game on quizlet. (http://quizlet.com/690390/scatter/). The initial time to beat was 30 seconds. Person with the best time got a I.O.U. We spent some time on quizlet, going over how to use it to study.
Then, Mr. O’Brien gave us some Unit Circle sheets for us to fill in, that when done, should look a bit like this:



















Then, we went over a few more formulas:





























We then covered linear speed and angular speed.
Linear speed is how fast the arc length is changing.
It can be expressed as:






And it is measured in terms of:




Angular speed is how fast the angle is changing.
It is expressed as:


(angle measure in radians)




And measured in terms of:




And finally we took a look at this ‘nice connection’ of the formulas:



















Next scribe was Lange.
Lange named scribe for Monday 2/8/10 as Anna.

February 4, 2010

Class started off today with a homework quiz, which we had twenty minutes to do; it was based on the homework problems that we have had for the past couple of days.

Then we went over homework questions from last night: page 291, #'s 71-93 odd, 100-106, & 108. We had questions with 103, 93, 75, & 73b.

Question 103 had to do with angular and linear speed. The formula for angular speed is , & the formula for linear speed is . Question 73 had to do with the minutes & seconds of already tiny degrees. Mr. O'Brien said to do it with this kind of format:

330˚ 25' = 330 + = 330 +

Which looks a little complicated, but actually makes sense. You basically just take the minutes & divide them by sixty to get the ratio (since there are sixty minutes in a degree), & then divide that by sixty again.

For 75, which was a little of the same but in a different order, you had to convert a degree with a decimal to a degree with its minutes. This one was 240.6˚, which we converted to 240˚ 36' by multiply the .6 decimal by 60 minutes; this gave us 36', which was the exact number we needed. Yay!

We moved on to the six trig functions next.. Which is kind of a big deal!



This is an example of a filled out trig circle, which will be referenced a lot in this post. Important symbols to know are:
r = radius
= radians
s = arc length

There's a quizlet on the blog that goes over the six functions of trig that you could find helpful. However, an easy way to remember these is SohCahToa.

y = a, r = c, x = b
Sin = y/r, Opposite over Hypotenuse, S-O/H
Cos = x/r, Adjacent over Hypotenuse, C-A/H
Tan = y/x, Opposite over Adjacent, T-O/A

Then we talked about the reciprocals of these equations.
Csc = r/y
Sec = r/x
Cot = x/y

As you can see, these angles are the inverses of their respective functions.

Then we went over the Japanese trig function applet, which allows you to input any degree for x, & it will output that angle in standard position, from which you can establish the sine of that degree. We did a little experimentation with the applet, & found out that in a UNIT CIRCLE (a circle whose radius is one):
Sin = y
Cos = x
Tan = sin/cos

CRAZY.

This means that:
Csc = 1/y
Sec = 1/x
Cot = x/y

Which was SO bizarre, because that meant that to find the cosine or sine of ANY unit circle, all you had to do was find the coordinates of the value & then choose either x or y for cosine & sine respectively.

EX: Find the sin45˚.
The coordinates on the unit circle for 45˚ is . So, since sin = y, the sin45˚ = . Which works!

You can do the same idea with the cosine, just use the x value of the coordinate rather than the y. A good site to help memorize the coordinates of these values (ew) is DudeFree.com, or the multitude of quizlets down below this post. I'm sure this is how everyone is going to spend SuperBowl Weekend, because this is so much more fun than anything else I can think of :D ... Just joking!! But seriously, this is a good idea to do at some point, even if it's just a half hour of studying.

Have a good weekend everyone! Anna, have fun scribing!

Links for Thursday, Feb. 4

Quizlet.com for the common radian-degree conversion values warm up

six trig function definitions in quizlet.com

sine function of any angle applet
cosine function of any angle applet
our unit circle applet

HW:

• p. 299/1-51 odd
• Keep working on your memorization. Here are some more resources you can use:

1. Quizlet.com for 32 trig basics flashcards
2. Quizlet.com for 98 flash cards of the common trig values
3. Quizlet.com for all 198 common trig values!

1/29/2010 - Intro to Trig

We started class by reviewing the “Francois and his Pedometer” packet. While we were going over the packet, we learned some new symbols that were introduced to us in #9-12. (E & Z). In trigonometry, the common variable is k. K is always defined as K E Z. This means that K is an element of Z, which is the set of all integers.

Next, we did an activity on Geogebra. We created angles, and manipulated them to learn about negative angles, and angles over 360 º.

















Ex. Negative angle (Left)
Ex. Angle > 360 º (Right)

One thing that we noted about the angles that we made was that when the initial ray of the angle was manipulated and moved around, the Arc length and Radius changed, but the radian’s value did not. When the terminal ray was moved the Arc length and the Radian value changed. These changes were also noted as proportional.

We also looked at a new packet of notes titled, “Introduction to Trigonometry Notes”. From the packet we learned:
The radian measure of one revolution is approximately 6.28 (2π)
The radian measure of one half of a revolution is approximately 3.14 (π)
Conversions between radians and degrees
Conversions of common radian values to degrees

Our homework for next class was practice problems in the book. Mr. O-Brien said that it was okay to only do a few from each section, if you felt that you understood the concept. Also, there are flashcards on Quizlet.com on common radian degree conversions.

Next Scribe: Julia