About Trig.... Some interesting links

Histoy, importance, and definition of trig:


http://www.bookrags.com/research/trigonometry-wom/



Summary of a bunch of trig:

http://www.bookrags.com/research/trigonometry-wom/


This is interesting, however it's mostly stuff which we have in our book or that Jojo has told us:

http://www.physicsforums.com/showthread.php?t=290472

Trig Cheat Sheet

I thought this might help y'all study for the test and even the final!

http://tutorial.math.lamar.edu/pdf/Trig_Cheat_Sheet.pdf

Cliffnotes 4 MATH?!?!?!?! (Fundamental Identities)

Yes everyone I know....I didn't really believe it either but Cliffnotes.com not only helps you with you're lit assignment that you didn't read it can also gives brief notes on your math assignments that you didn't read or (since we wanna be optimistic here) that you didn't understand from the book . SOOO if you're a tired student athlete that has no time or energy to study for your OH SO IMPORTANT DQ then Cliffnotes.com may be able to help you get a brief understanding before class.


http://www.cliffsnotes.com/study_guide/Fundamental-Identities.topicArticleId-11658,articleId-11608.html

(P.S. The colors hold no meaning what so ever)

Idnetities

This is a list of the identities from chapter 3.2. This should be helpful to those of you who are struggling with remembering all of them (including me...) Hope this helps!

Tutorial Team for Upcoming Test

Hey guys! So Jojo asked to post the next tutorial team. It has to be done by Tuesday March 30 by 9:00 pm. So get together and share it with Jojo. Good Luck! (Chapters 2.1, 3.1-3.4)

-Isabel

-Madison

-Brian

-Lizzie C.

March 25th, 2010

Hey Everybody,

I hope you all enjoyed your math class today and that there were many breakthroughs!

Well, to recap....

First we completed a daily quiz, number 20. The quiz involved simplifying equations using identities.

Next, we went over section 3.2 by doing an example:

2tan^2x = (1/csc x - 1) - (1/csc x + 1)

one way to solve an equation involving the subtraction of two equations is by multiplying the denominators to either side like so:

[1 (csc x +1) / csc x - 1 (csc x + 1)] - [1 (csc x - 1) /csc x +1 (csc x - 1)

continue to solve:

[(csc x + 1) - (csc x -1)] / csc^2 x - 1

= csc x + 1 - csc x +1 / csc^2 x - 1

= 2 / csc^2 x - 1

= 2 / cot^2 x

= 2 tan^2 x

Finally, we all ended class by dividing up into small groups to discuss our homework, and work through the problems we found difficult. During this time, Jojo walked around and helped those of us that still had questions.

Although many people found the work frustrating, Jojo helped us work through and understand the problems better, and we were very grateful for that and proud that we understood the material much better.

Don't forget to study for the test coming up next Friday, and also catch up on homework, flickr, corrections and contributions on the blog etc.

You can do it!!!

Good Luck,

Emma :)

PS. the next scribe poster is....Brian!!

Scribe Post 3/24

Hello Everybody,

Today we went over more problems from 3.2, namely verifying and identifying rational identities.

Rational Identity: Trigonometric expressions that are equal fractions set on either side of an equal sign.

here's an example from page 177 in your trig book

cos(a)/(1-sin(a)) = (1+sin(a))/ cos (a) where (a) equals alpha

(cos(a)/1-sin(a)) x (1+ sin (a)/ (1+ sin(a)) = (1+sin(a))/ cos (a) x (1+sin(a))/ (1+sin(a))

Multiply both sides with either the numerator or denominator of the opposite side (in this case numerator of left side) to get like terms

cos (a) (1+sin (a)/ cos^2(a)= (1+sin(a))/ cos (a) Here you use your knowledge of the Fundamental Pythagorean Identity {sin^2(a) + cos^2 (a)=1 } to solve (all trig identities are in the very last page of our textbook)

(1+sin(a))/cos(a) = (1+sin(a))/ cos(a) YAY IT WORKED!

We also did example 5 on page 177 in your textbook. The main difference with that problem is that you have to know to split up the fraction identity:

(csc(x)-sin(x))/ sin (x) becomes csc(x)/sin(x) - sin(x)/sin(x)

Here's a youtube video of a guy doing various different types of identity problems(specifically identity problems with fractions like above).

http://www.youtube.com/watch?v=OJz-fOzFbEc&feature=PlayList&p=2AFAB7497192607A&index=19&playnext=2&playnext_from=PL

JoJo would like to remind everyone that he is in his room Wednesday and Thursday at lunch and break to help anyone who needs it. Remember to get your Flickr, and Pbworks stuff in as that's being checked as we speak. Get help now, we have a TEST next Friday.

Next scribe is Emma

Really Helpful Video!

Hey guys - I found another video to help us out.
This one defines all the identities and explains how to simplify an Identity.
This isn't exactly what we were doing in our homework, but it's very similar, and should be very helpful.

Enjoy!

http://www.youtube.com/watch?v=P1JDbIldb_Q

(copy and paste link)

Sin and Cos function review

This site is a pretty good review, make sure to check out all of those links in the side, they're really helpful.

http://math2.org/math/algebra/functions/sincos/properties.htm

-Luke

scribe post-bekah-chapter 2 and 3 catch up

Monday, March 15th.

Today we went over sine and cosine in more detail.

Jojo wrote this equation on the board:

y=a sin (x-h) + k

if y=sin x

and we were supposed to figure out what a, k and h were.

h=0

k=0

a=1

We learned that the amplitude is the height of the degree axis to the highest part of the function. So pretty much, amplitude = the highest point

If you are given y = 2 sin x than the amplitude (a) = 2

The amplitute would also = 2 if we were given y = -2 sin x, because the highest point on the degree axis would still - 2

We than began going more into the amplitude, with more depth.

We figured out that a = a positive number if the line on the graph goes up from the origin.

We then figure out that a = a negative number if the line on the graph goes down from the origin.

We started vaguely going over the other parts of the equation (y=a sin (x-h) + k) that Jojo gave us earlier.

From this we learned that h decreases if it translates horizontally (left or right.)

Jojo told us that the difference between sin(x) and sin (x-0.6) is that one the graph it shifts differently.

-->it's all about the shift.

When h = a negative value (x-h) = (x-(-h)) so it = (x+h) shifting to the left.

We also learned that h is done in radians.

Tuesday, March 16th.

We first revisited the AMPLITUDE, and clarified that you can find the value of the amplitude by the distance between the x-value (0,0) to the highest point (on the y-axis) or the lower point (on the y-axis.) You measures the midpoint to the highest or lowest y-values. We learned the equation you can use to find the amplitude:

the absolute value of 1/2(max y-value - min y-value)

We then learned what the Sine and the Cosine Graphs look like.

y = sin(x) the graph goes through the center origin (0,0). Going up (on the right) one unit up, one then

one unit down, then one unit down again, then one unit up, then one unit, then one unit

down, than one unit down. The left side does this same thing, but first it goes down and then

goes up and than goes up and than goes down, etc.,

(You should be seeing a pattern on the graph in your head)

-We also learned that another way of finding out what the graph looks like is by using your calculator.

You press the button that has: y= :on it. You then plug in whatever you are given. Then you press

the button that says: GRAPH :on it. There you will have the graph for your sine or cosine equation.

y = cos(x) the graph has the same pattern, except it starts on amplitude of the positive part of the y-axis therefor

going down symmetrically on each sides.

-You can also use the calculator to figure out the graph for the cosine equation.

Wednesday, March 17th.

We learned went over some main Identities today and then practice some problems.

The first identity we learned about was the:

Reciprocal Identity: sin x = 1/csc x csc x = 1/sin x

cos x = 1/sec x sec x = 1/cos x

tan x = 1/cot x cot x = 1/tan x

Jojo then showed us what tanx and cot x are in terms of sine (sin x ) and cosine (cos x):

tan x = sin x/cos x

cot x = cos x/sin x

The second identities we learned about were the :

PT identities: sin^2 x + cos^2 x = 1 --> divide everything by cos sin^2 x and you get

1 + cot ^2 x = csc ^2 x --> cot x (this leads back what Jojo taught us about the cos x)

sin^2 x + cos^2 x = 1 --> divide everything by cos ^2 x and you get

tan ^2 x + 1 = sex^2 x --> (this leads back to what Jojo taught us about the tan x)

Pretty much everything we learned upto here, today has been been the identites that lead back to the sine and cosine.

We then walked through problems 1, 2 3, 4 and 5 on Page 173 in our Trig Textbooks)

-During problem 5 we ran into FOIL-ing, which is a math technique that most of us learned from Algebra. To FOIL you foil down the problem:

(a - b)(a+b) --> you foil the equation. a * a, a * b, -b * a, -b * b, which gives you a^2 - b^2

Thursday, March 18th.

Today we learned how to find the coordinates on sine waves.

Friday, March 19th.

Today Jojo was ready to teach us about EVEN and ODD Functions:

On the board he had two sections of information written down:

1.) EVEN--> symmetric about the y-axis

cos(-x) = cos(x)

sex(-x) = sex(x)

2.) ODD--> symmetric about the origin

sin(-x) = -sin(x)

csc(-x) = -csc(x)

tan(-x) = -tan(x)

cot(-x) = -cot(x)

We than went over some examples from the book: Page 173 problems 24 and 26. We walked through each step of every problem.

24. sec(-x) - sec(x) -->we have to solve and we know/see that in normal math the answer would = 0 and that is completely correct. The answer is 0

25. cos(y) + cos(-y) -->this turns into

cos(y) + cos(y) --> which equals

2cos(y)

We than learned a few random things about EVEN and ODD functions.

-if you fold an EVEN function graph: f(-x) in half the sides will line up perfectly. The graph will have the same y-values on the positive and negative sides of the origin.

-EVEN = f(x) = cos(x)

_ODD = f(x) = sin(x)

March 22nd
Today we went had a daily quiz and then began learning about STRATEGIES in trigonometry.

Jojo states that "it's really a crazy puzzle but there's a method to the madness" -which was exactly what we began doing during class.

Jojo first showed us the strategy that uses:

VERIFYING IDENTITIES: 1 + sex(x) sin (x) tan (x) = sex^2 (x) -->(this is the identity and now you must verify it)

Step 1: Start with the most complex side and work towards the easier side.

1 + (1/cos(x)) (sin (x))(sin(x)/cos(x)) --> here we turned everything into sine and cosine)

Step 2: Divide everything by the cosine --> sin^2(x) + cos^2(x) = 1

1 + (sin^2(x)/cos^2(x)) --> 1 + tan ^2(x) --> sec^2

-If you didn't notice there are a lot of Reciprocal and PT identities--this shows us how important it is to really have the identities down-

We then went over Identifying the Factor for the different of squares:

The first thing we covered in this section was:

FOILING: any time you have (a^2 - b^2) it also equals (a - b) (a + b)

The second thing we covered in this section was:

PERFECT SQUARE BI or TRINOMIALS: (we couldn't figure out if it was bi or tri)

(a + b)^2 always simplifies to a^2 + 2ab + b^2

We were about to begin figuring out the RATION EXPRESSION but it was time for Monday Morning Meeting.

I am terribly sorry for this scribe hold up, i promise never to do it again. I hope that my scribes from last week are helpful to anyone who may need it.

The next scribe is..zoe

PoP -Trig Visualizations in "On to The Next One"

Yes, to all of you haters...there is trig in the Jay-Z video.  Comment specifically on the trig visualizations in the music video. No repeat visualizations please!  Good Luck!

Difference of Two Squares

http://themathpage.com/Alg/difference-two-squares.htm

This slipped my mind, and I couldn't find it in my notes (not a good sign, I know) and this helped alot... This is for all of those who are in a similar situation... :)


-Luke

Trigonometric identities

Trigonometric identities

Square Dancing - Opinionator Blog - NYTimes.com

Square Dancing - Opinionator Blog - NYTimes.com

Graphing Trig Functions - extra help

Here is a great website if you're still having some trouble graphing trig functions!

http://www.themathpage.com/atrig/graphs-trig.htm

(copy and paste link)

Good Review

Trig Functions

Hey this site gives a little bit more insight into trig functions and how they work. It's not the greatest but hey for those visual learners that learn solely from the notes of others this may help. Sometimes it's not the actual math it's how it was explained to you that can make it seem difficult.

http://sosmath.com/trig/Trig2/trig2/trig2.html

Just somethign fun

http://www.piday.org/million.php

Happy pi day!!!

I wonder how much more exact you can get estimating pi...

Helpful Definitions of Trigonometric Functions

found this over the weekend, it has information about nearly everything we've gone over for Trig so far this year.

http://www.knowledgerush.com/kr/encyclopedia/Cosine/

Exploring Sine Curves, by Kristina Dunbar, UGA

http://jwilson.coe.uga.edu/EMAT6680/Dunbar/Assignment1/sine_curves_KD.html


Found that while looking for pics for the weekend assignment... Thought it might help/interest someone.

Double PoP (worth two PoPs) -Today is Pi Day!

“Imagination is More Important Than Knowledge”—Albert Einstein

1. What wouldn't you be able to do everyday without (Pi)?
-I wouldn't be able to drive to work (aw man, now you can't use driving as an example! :-) ).

2. Have an imagination and be creative! Design a digital design or sketch of the word "Paideia" that incorporates symbols, signs, and formulas we have used so far in this class! (Art should not be larger than 8 x 11 or the computer screen...)
See the Google Logo example above.

All responses and drawings due 5pm Thursday.

Here's a great video on graphing functions!

http://www.mathtv.com/index.php?playlist=13371

www.mathtv.com

Presented by MathTV.com

Scribe Post: March 11th 2010

DO NOT READ UNLESS YOU HAVE TAKEN THE TEST!!!

...SERIOUSLY. =)

Today in class we started going over section 2.1.

This section of the book includes:



The definition of a Unit Circle, that r (the distance to the origin) = l

Picture of a Unit Circle:


Using our knowledge of the Unit Circle, we learned more about the Trigonometric Functions.
In the first part of this section, the book showed you how to solve a function without a calculator, just like what we had to do on the first ten questions on the test yesterday.

This box (as found in the book) was extremely helpful in figuring out how to do this:
'If alpha is an angle in standard position whose terminal side intersects the unit circle at point (x,y), then
sin alpha = y cos alpha = x tan alpha = y/x csc alpha = 1/y sec alpha = 1/x cot alpha = x/y provided that no denominator is zero.'

So all you have to do is take the points given to you on the unit circle in coordination with the degree or radian given, and use the x &/or y according to the function.

Example: sin 45 degrees

The answer to this is the square root of two divided by two because that is the y point that the unit circle gives you.

if the problem was csc 45 degrees, than the answer would be 1/ square root of two divided by two.



We also started going over graphing trigonometric functions. We learned that when graphing the functions in an xy-coordinate system (normal graph) you use x as the independent variable and y as the dependent variable. So, for example, if sin (x)= y, and you used 30 degrees as your x, than sin (30 degrees)= 1/2. x, 30 degrees, is the independent variable (or input) and y, 1/2, is the dependent variable (or output). Another example JoJo used in class was if you used day light or moon light and time Proxy-Connection: keep-alive
Cache-Control: max-age=0

day. In this the time of day would be your imput, and your output would be the amount of sunlight or moonlight there is left. This only works with things that move within a rotation, like the unit circle.

Hope this helps you guys understand!

Tomorrow's scribe is Rebekah. =)

Aristotle and Math

So the other day before class I read some Aristotle (thinking about what courses to take next year, decided to do some pre-reading) and as I grappled with the first two paragraphs of his book on politics, I got the feeling of turning over a concept in your mind. A feeling that is almost pain, and almost pleasure for me. It's when you are beginning to understand something but cannot even begin to grasp or appreciate it's magnitude and importance in it's entirety.
Because of my reading Aristotle, I was able to grasp concepts with much more ease than I could before--because my brain was already working in a mathematical way (more on how philosophy is essentially math later). A few things occurred to me that I found very interesting that I'd like to share with all of you:
1. At a very young age we already understand complicated and trigonometric principles: we understand what one is. We understand having one which to us seems very simple, however in that we also understand trigonometric principles and functions without even realizing it:
sin^2(alpha) + cos^2(alpha) = 1. Inherent in one, is the fundamental identity of trig, the Pythagorean Theorem, angle measurements, radians, degrees, minutes, seconds, an entire world of trig all inherent in the simple number which we grasp at a very young age... One.
2. Math is essentially philosophy. Just look to the great philosophers, Aristotle wrote entire books on what he called, "logic." And "logic" is a common word used to refer to mathematics. Many great philosophers were also mathematicians who discovered groundbreaking theorems and proofs, such as Pythagoras. Also, what is math other than the pursuit of knowledge to better understand the world we live in? And what is philosophy other than this? Why do we seek knowledge other than to know what to do with our lives, and how the world around us works? Philosophy is essentially the same thing: figuring out what's ethical and what is not, how states should run, how people should live, free will or predetermination, the meaning of life even. Also, I was talking to Cullen a teacher here at Paideia who teaches philosophy, and he said that when he was studying philosophy he had to take a class that was all proofs, and that inherent in algebra and in such things as proofs is philosophy in the case that they make an argument for something or other...

Anyways, just food for thought...

-Luke

Section 1.4 Helper

Hey Guys,

I realize this is late, but if your up studying or check the blog early; this is very helpful. Look at this table. In section 1.4 the majority of degrees someone will get if they used the reference angle to find the exact value of a degree or radian will be either: 30, 45, or 60. Try it out in any of the problems we were assigned on that section and you will see. Now, most of the time we are told to set up a standard 45-45-90 triangle or 30-60-90 triangle; you can avoid ALL OF THAT if you memorize the Unit Circle Values for Sin, Cos, and Tan; since you can get their reciprocals: Cosecant, Secant, and Coterminal by for e.g. Cosecent = 1/Sin(x), or Secant = 1/ Cos(x), etc.

I hope this helps.

http://web.gccaz.edu/~jgutier5/Tables/The%20Unit%20Circle%20Table%20Of%20Values.pdf

45 45 90 triangle's and relation to unit circle

The 45 45 90 triangle is useful in determining the ratio of side to side, and side to hypotenuse. any time the two sides of a triangle are exactly the same length, the hypotenuse will be the length of the side times the square root of 2. this make sense because 45 degrees is exactly half way between 0 and 90 degrees.

PB Works Answer Manual -Back in Action!

Our Chapter 1 Answer Manual is back in action. You may need to give the link a second when you click on it, but it should work now!

If you are having trouble you can submit answers in the Chapter 1 Answer Manual Back Up!

Scrib Post (Reference Angles and 1.6 Intro)

Hi JoJo's math students!!
Today was a very productive day in class. We covered reference angles(like our Thursday night homework) and an intro to 1.6.
I reference angle is the positive acute angle (aka theta prime) created by the terminal side and the negative or positive theta on the x-axis.

*The picture of the left is an angle in standard position. With the initial side (first) and the terminal side (end).







This is what the reference angle looks like.(reference angle in red)

You can solve reference angles by creating a 30-60-90 triangle (like the example we did in class today.

So if you found out that your reference angle was 60 degrees you can form the triangle on the initial side and solve. So you would have sin= y/r so
sin 60= square root 3 over 2





In case you did not know, this is what your 30 60 90 triangle will look like!!





Now the other part of class was an introduction to 1.6


SOH
sin theta= opposite/hypotenuse


CAH
cos theta= adjacent/hypotenuse

TOA
tan theta=opposite/Adjacent

So say your problem gives you the hypotenuse. You could use the SIN to find the Y and the COS to find the x. Tan is great to use it you are trying to find a other degree or (alpha)!!

I hope this was helpful!! Have a great weekend!!
Jocelyn

The next scribe is Katherine

Section 1.5 Identities and Reference angles

Today we discussed Section 1.5 (starting on page 81 in the text book). This section is all about identities, and reference angles, but we focused on identities today.

An identity is an equation that is satisfied for all values of the variable for which both sides are defined. Some examples are: x+x=2x, and x/x=1.

The Fundamental identity for any and all real numbers is: Sin² α+cos²α=1 but we can rearrange this identity into two other equations to make life a little bit easier.

cosα=±√(1-sin²α)

sinα=±√(1-cos²α)

If you're trying to find cosine and you are given the value of sine, try plugging it into the first equation or vice versa. The important thing to keep in mind here is that you will be told the quadrant in which the angle lies. This is important when you are deciding whether the value of sine or cosine is positive versus negative. Here's a handy little chart on that!

When you don't have this handy dandy chart, think about it logically: r is the distance from (x,y) to the origin, so r has to be positive (distances can't be negative). Then if x or y is negative. Sine, cosine, secant, and cosecant will also be negative, because a positive multiplied or divided by a negative is a negative.

I hope this has all made sense! Please let me know if you have questions!

See you tomorrow,

Elizabeth

Scribe Post, March 3 2010, Marc

Hey everyone,

Before I start, I must apologize for posting this so late, I only just got to a computer.

Today in class, we did more review on trig functions. Jojo worked through a few of the homework problems with us. We did not use calculators, but instead found the exact values of each problem we worked out. The method my class period used most was the unit circle method, where you use the unit circle to find the values of , , and .

For example:

To solve this, you must first use the chant Jojo taught us, 30, 15, 15, 30, to find the location of the angle in the unit circle. http://www.mathpeer.com/images/trig/unit_circle.gif That is a link to a picture of a unit circle. If you have one of these on hand, you can simply look at it and find 135 degrees, which is halfway between and . The given angle is .

Jojo told us that an easier way to work with any angle would be to treat it as if it were in the first qudrant. So in the first quadrant becomes . To find , start by forming a right triangle with the radius used to form the angle and the horizontal axis. This will form a 45-45-90 triangle, which means that the two legs will equal 1 unit, and the hypotenuse, which is the radius, will equal . Now, is a function, which is equal to . So . You cannot leave in the denominator, because it is a rational number, so multiply both the numerator and the denominator by . This will leave you with , which multiplies out to be ., if it is in the first quadrant. Now, a 135 degree angle will be in the second quadrant. Because a equation will only deal with y-coordinates, the answer will not become negative because the y-coordinates in the second quadrant wil lbe positive.

I do hope all of this made sense.

As a side note before I post: In case anyone has not posted this link, http://www.codecogs.com/components/equationeditor/equationeditor.php will take you to an HTML equation editor, which is quite useful for posting on blogspot as at the top of the page (at least when i am posting) there is the option to edit HTML. Simply type the desired queation, and it wil ltranslate the equation into HTML code. Then cpoy and past this code into the desired place under "Edit HTML" part of the post, and the equation will appear in the text. This sounds confusing, but its simple once you get the hang of it. Hope this helps.

Marc

Visual Learners...

This website has a lot of simple but informative videos on how to do basically everything we're learning about trig functions (sin, cos, tan, csc, sec, and cot). There are usually a few videos per subject lesson, so if one person is too annoying for you or you just aren't getting it, you have other options.

This site also has other categories; it basically covers every math topic out there in easy to follow vidz. Enjoy :)

http://www.mathtv.com/

Chile Quake Changes Time...

The earthquake in Chile was so strong that it shifted the earth's axis, making all days from here on out 1.6 microseconds shorter... hppt://www.businessweek.com/news/2010-03-01/chilean-quake-likely-shifted-earth-s-axis-nasa-scientist-says.html

Trig Help

Hey Guys!
Here's a really helpful website to help with anything that's confusing.
It has great info on trig ratios, trig functions and so on.
Enjoy!

http://www.trigonometry-help.net/
(copy and paste)

Flashcards about Trig Functions

Flashcards about Trig Functions