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

Using identities (sum and difference examples)

Here are my two examples, i wasnt sure of what i was doing, but
βεθany was nice enough to help me out!!


Example 1:

Simplify: sin6x sin4x - cos6x cos4x

(for this you will have to use the identity) :
cos(α+β)= cosαcosβ - sinαsinβ

sin6x sin4x - cos6x cos4x =
-(cos6xcos4x - sin6xsin4x) = -cos(6x+4x) =
-cos10x

Example 2:
Determine the exact value of sin 75°

To begin, use the identity: sin(α+β) = sinαcosβ

sin(30+45) =sin 30cos 45 + cos 30 sin 45

= 1/2(√2/2) + √3/2(√2/2)

=√2/4 + √6/4

=√6+√2 /4

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!!