3rd Hour Honors Algebra 2 B (Spring 2012)

Tuesday, April 24, 2012

Section 7.3: The Addition & Subtraction Formulas

In this section, we derived the six addition and subtraction formulas for sine, cosine, and tangent.

We started by finding the addition identity for sine, beginning with a triangle.

∆DFB: ∆ABC: ∆ABD:

cos x = DF/DB sin x = BC/AB sin y = DB/AD

DFL DB cos x BC = AB sin x DB = ADsin y

cos y = AB/AD

AB = AD cos y

So, we used these findings to figure out the sine of x + y (angle A). To find this, we started with:

Through substitution, we performed the following steps:

We then began to substitute more into the derivation:

By canceling out the AD, we end with:

sin(x+y) = sinycosx + cosxsiny

Here are all six of the addition and subtraction identities:

Sorry for my messy handwriting :)

These are all of the identities, found in similar ways as the sin addition identity. These are true for all values you plug in, because they are identities.

I think that's about it for 7.3, see you all tomorrow!

- Jessica

Tuesday, April 10, 2012

lesson 6.5

To simplify the concept with the sine function, we often use the formula y=a sin [b(x + c)]+d

we can use the same variables for the cosine function y=a cos [b(x + c)]+d

Below are the graphs of y=sinx and y=2sinx

as you can see, the a variable from above designates the amplitude of the graph

these are the graphs of y=sinx and y=sin2x

in this example, you can see than increasing the b variable shortens the period of the graph.

changing the c variable shifts the graph on the x axis a value of -c

this is called phase shift

changing the d variable vertically shifts the graph a distance equal to d

this is called mid-line shift

the graph for cosine behaves very similarly to the sin graph, however the cosine graph reaches its amplitude (1) when x=0. other than this shift, the two graphs are identical. the cosine function is even, while the sine function is odd.

Thursday, March 29, 2012

6.5 Trigonometric Graphs

TRIGONOMETRIC GRAPHS

Graphing

'a' leads to vertical stretching. Vertical stretching will cause a change in amplitude.
'b' leads to horizontal compressing. Horizontal compressing will change the period.
'c' leads to shifting left or right. This left or right shifting is called a phase shift.
'd' leads to shifting up or down. This up or down shift is called a midline shift because it will affect the midline.

VERTICAL STRETCHING

compared to

AMPLITUDE/PERIOD

HORIZONTAL COMPRESSING

compared to

L/R SHIFT (PHASE SHIFT)

compared to

The graph on the left is showing a left phase shift. The graph on the right shows a right phase shift.

U/D SHIFT (MIDLINE SHIFT)

This graph shows an upward midline shift.

This graph shows a downward midline shift.

MIDLINE

This graph shows

in the thinner and wavy lining and their midline's in a thick line through the middle of each.

And that's about it, other than The Nonagon Song, of course.

Alex Hackert

Sunday, March 25, 2012

6.2 Part 2 Identities

Identity: An equation which is true regardless of what values are substituted for any variables.

Identities apply to basic algebra as well as trigonometry, and especially to what we are learning. Identities apply to sin, cos, and tan. Because they each have a reciprocal we have learned about( csc, sec, and cot), its quite simple!

These are reciprocal identities. There are also Quotient Identities. These are derived from the 3 basic trigonometric functions, using substitution based on their definitions:

and

Finally, in class Mr. Wilhelm helped us derive the 1st of 3 Pythagorean Identities, using the Pythagorean Theorem, and using substitution by definitions:

Becomes...

To solve an identity, you must change and work with one side without altering the other. As Mr. Wilhelm told us, it is best to work with the crazy messy side and try to work towards the nice clean one. Lots of substituting!

Soo thats basically it i think, except for :

http://www.youtube.com/watch?v=3nG1ckY7thw

Please Mr. Wilhelm, I've never seen it and I think to fully enjoy it we need to be all together in class as to create the right atmosphere. :)

Good luck!

~Kendall

Saturday, March 24, 2012

6.3 Graphs, Periods, and Even and Odd Functions

Why hello there! This is Nonny the Nonagon stopping by to help everyone with section 6.3. His good friend, Sergei the Circle has decided to come with him to help. Sergei is really good with explaining the graph of sin.

Now, Mr. Wilhelm told us how to look at a graph of sin. Although he had a fancy website, here's a picture showing how a point on a circle coincide with the sin of a point. (This was Sergei's favorite example)

When you keep going around a circle to show the graph on sin, that wave, formally called a period, will keep repeating. A period is the smallest part of the wavelength on the graph before it repeats.

Mr. Wilhelm also told us these two equations which go along with the graph below when:

Basically, he's just showing us how the two angles are both solutions to the equation, although they are: in completely different quadrants and two different standard position angle measures.

Flashback to Honors Algebra 2 A, we went over functions today in class. Though we were a little rusty on them, we ended up figuring out how this section relates back. But before we get into it, lets review what even and odd functions are.

Now, is there any difference between those functions and these?

The answer to that would be a no. All we did is change the x to a theta so we can use these equations for circles. But which trigonometric function goes with which? Good question.