Pi Day!!!

HAPPY PI DAY EVERYONE!!!

picture via the Hope University math newsletter Off on a Tangent

In case you weren't aware, it is also Albert Einstein's birthday. Also, a proper Pi Day celbration should not begin until it is 1:59:26 on March 14 (π = 3.1415926... = 3rd month, 14th day on 1:59:26).

Today's lecture was short as Mr. Maksymchuk decided that we should use the period to work on the pre-test and catch up with accelerated math. Employing the sum and difference identities that were taught to us yesterday, Maksymchuk showed us how to derive sleek-and-sexy numerical values for trigonmetric functions with"strange" angles, among other things:

Here he derived an equivalent expression for sec(θ-π/4):

Continuing with the sleek-and-sexy values:

And finally we were given our assignment:

Mr. Maksymchuk finished this around 12:30 p.m., giving us about 40 minutes to work on our pre-test and finish up our accelerated math.

Anyway, since this post was so short I think I will provide my fellow classmates with a puzzle they may or may not choose to figure out:

You have 9 balls - all of which are indistinguishable by sight - and a balance scale. Eight of the balls weigh exactly the same while the last one is either heavier or lighter than the other eight. This differential in weight is so small that you can not tell by holding the "odd" ball in your hand whether or not it weighs any differently than the other eight; that is, you will have to employ the balance scale to figure this out. What is the smallest amount of weighings that you can do to find the odd-ball-out? How did you carry these weighings out?

Feel free to post your solutions.

-IB

Idenities Example

I have decided to use an example of one Double Angle Identity, and on example of a Sum and Difference Identity..

1) Find the exact value of the following:
a. sin (π/12) =sin (π/3-π/4) = sin(π/3) cos(π/4) - cos(π/3) sin(π/4)
= (√3/2)(√2/2) - (1/2)(√2/2)
= (√6/4) - (√2/4)
final answer = (√6 - √4)/4

2) Solve for "x" where 0° ≤ x ≤ 360°

a. sin2(x) = sin(x)

2sin(x) cos(x) = sin(x)
2sin(x) cos(x) - sin(x) = 0
sin(x) (2cos(x) - 1) = 0

sin(x) = 0, therefore x = 0, 180, 360

2cos(x) - 1 = 0

cos (x) = 1/2, therefore x = 60, 300

Trigonometric Sum and Difference Example Post

1). Find sin (60º-45º)

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

sin(60º-45º) = sin(60º)cos(45º) - sin(45º)cos(60º)

=(√(3)/2)(√(2)/2) - (√(2)/2)(1/2)

=(√6 - √2)/4

2). Find cos(-x)

Employing cos(2π-x) = cos(-x) and cos(a-b) = cos(a)cos(b) - sin(a)sin(b) we have:

cos(-x) = cos(2π-x)

= cos(2π)cos(x) - sin(2π)sin(x)

= (1)cos(x) - (0)sin(x)

= cos(x)

Voilà!

Sum and Difference Identity Examples

Well, I basically just copied what the examples were in the green Mickelson book, but changed the numbers, but just working through the examples and my own really helped me understand them better. Here's what I came up with. By the way, I'm sure there's a way to use greek letters, but I couldn't find it so I'm substituting a for alpha and b for beta.

Example 1

Simplify: sin4xcos3x - cos4xsin3x

By using the identity sin(a - b) = sinacosb - cosbsina,

sin4xcos3x - cos4xsin3x = sin(4x - 3x)
= sin(x)

Example 2

Determine the exact value of cos(315). (That would be 315 degrees, but I don't know how to get a degrees sign on here either.)

By using the identity cos(a - b) = cosacosb + sinasinb,

cos(360 - 45) = cos360cos45 + sin360sin45
= 1(SQRT(2)/2) + 0(SQRT(2)/2)
= SQRT(2)/2

OR

By using the identity cos(a + b) = cosacosb - sinasinb,

cos(270 + 45) = cos270cos45 - sin270sin45
= 0(SQRT(2)/2) - (-1)(SQRT(2)/2)
= -(-1SQRT(2)/2)
= SQRT(2)/2

Hopefully that makes some sort of sense. Don't forget, tommorow's pi day!!

Wednesday, March 12

Today in class we continued to learn about triginomic identities and how to solve them. To prove an identity to be true, basically you must try to make the left hand side of the equation equal to the right side. Eight basic techniques to try and solve or simplify an equation to prove its identity are shown below (a-h).

If the left hand side (LHS) equals the right hand side (RHS) then the identity is proved to be true. The correct terminology is shown in the slide below
In order to master the art of proving identities you must practice practice practice! To some, (like myself) it can be very difficult and frustrating which is why it is important to keep up with your exercises and practice.It is important to know that to verify and to prove are similar but not the same. When you are verifying something you finding an identity for a specific case but it does not prove the identity for the whole equation.






Similarily to graphing, verifying cannot be used to prove an identity but you can use it to disprove.

Together we went through a few different problems for proving identities which are shown in the above slides.

Even though today was our deadline for accelerated math, Mr. Max was ever so kind to give us until friday to get done our first nine objectives. So don't forget to get those done by 3:40 friday and for tomorrow have questions 1-9 of exercise 15 to complete.

Thats all for now, so remember, keep your eye on the ball!

By the way happy birthday Cindy Lou!!! Hope you have a good one!

Picture courtesy of "http://www.freerangechick.com/images/bdayCard/BBigBirthdayCake.jpg"

Tuesday, March 11

We began the class by discussing Accelerated Math. The deadline to master 9 objectives has been moved to the end of class on Friday. Also on Friday, we will be given a pre-test to take home in preparation for the test on Wednesday, March 19. The answer key for the pre-test will be made available on Monday.

Today we started learning about Trigonometric Identities. The following slide shows the definitions of a trigonometric function and a trigonometric identity. We learned that identities are a lot of logic and used a circle, triangle, and square to figure out some of the concepts of identities. We also used Graphmatica to check. It's important to remember that a graph can disprove an identity, but it cannot prove one.


The next two slides show some of the
common identities we will be using. Note that the first slide needs to be memorized, but the identities from the second slide are found on our formula sheet.


This slide shows an example using an identity to solve a problem. One thing to note is the rationalization of the answer. Mr. Max said that he prefers a rationalized answer, but we won't lose marks on the exam if they aren't. Just watch if he puts on the tests to rationalize our answers.


Here is an example of how to simplify an equation using identities.


The assignment from the cumulative exercises for tonight:

So you may have noticed the word nefarious seemingly randomly placed on one of the slides. Here's the dictionary.com definition: extremely wicked or villainous. It was used in reference to a plot that each of us are capable of concocting, but most certainly never would. Hopefully that clarifies for anyone else who didn't understand.

www.blazelabs.com/pics/donkey.gif

Again, I would appreciate questions and or comments about my post that would go towards making it better. Adios.