Friday, April 2, 2010

We spent a good deal of class working on the warmup problems. Basically, it was the application of the Fundamental Trigonometric Identities. The primary benefit of the exercise was that it gave us a chance on making our proof format more acceptable. Format work included using the equals sign in a transitive manner, and making sure everything was thoroughly explained. If you’re not evidently applying a property, it’s best to leave a side note for thorough explanation. For example,

If cotx + csc= cos/(sin^2), you couldn’t make the leap to “cos/(sin^2)=cos/(sin^2)” without first explaining your manipulation of cotx and cscx.

On concept that came up several times throughout the warm-up was the technique of multiplying by fufoos to get lowest common denominator. This can be particularly useful with complicated fractions.

After we had learned to write proofs in a more explanatory manner, we started work on the homework problems.
While working on the homework, there was some reviewing of how to derive the other pythagorean identities. For those who need a reminder, just divide the values on both sides by (sin^2)x or (cos^2)x. Basically, working on the homework solidified strategies we had previously gone over, like multiplying by fufoos.

Then, we worked on Solving Trig Functions. Instead of trial and error, we worked with technology to solve the equations. Note that some of the solutions to these trig functions looked similar to the format used in our Francois worksheet.

At the end of the class, we briefly talked about future algebraic solving.

Thursday, March 4 Class

We went over quiz, noting several things...

First, when Mr. O’brien says to graph both and angle and its reference angle in standard position, the initial sides of both angles must be on the x-axis. Second, much as sin(90º-θ)= cosθ, sec(90º-θ)= cscθ

After going over the quiz, we went on to discuss the inverse trig functions.

y=sin^-1x

y=cos^-1x

y=tan^-1x

Notation problems: sin-1x does not equal (sinx)-1

Because of this potential confusion, there are other notations to represent the inverse functions:

arcsin (x)
arccos (x)
arctan (x)

We saw that on grapher, arcsin(x) isn’t a function. On our graphing calculators we saw the inverse trig function is a portion of the entire function. We consider the inverse trig functions to be a collection of all the points possible while still being a function.
The same applies to Cosine, which doesn’t pass the horizontal line test any more than the Sine function does. A single Tangent wave passes the horizontal line test, and that is all that’s graphed of the arctan function.
Of course, all three graphs look different from the originals, since they are inverse functions, and therefore have reverse (x,y) coordinates.

Here are links to the three inverse functions. This should help make their domains and ranges pretty clear.

http://www.math.rutgers.edu/~greenfie/mill_courses/math151a/gifstuff/arcsin.gif

http://www.intmath.com/Analytic-trigonometry/arccosx.gif

http://upload.wikimedia.org/wikipedia/commons/f/f6/Arctan_plot_real.png


arcsin (x)
Domain: [-1, 1]
Range: [-π/2, π/2]

arccos (x)
Domain: [-1,1]
Range: [0, π]

arctan (x)
Domain: [all real numbers]
Range: [-π/2,π/2 ]

Use of inverse functions: We can plug in coordinate points along the unit circle, and the inverse function gives the angle. It gives angles in quadrants where that particular function (sin, cos, tan) is positive.

Then we took the quiz, which brought us to the end of class.

According to the tags on the side, Collin has only done one scribe post, so now it's his turn for next class.

Friday, October 23rd

We warmed up with two simple questions, one from the book and one that Mr. O’brien came up with. The problem from the book was number 42, a cubic function with three different roots. The other one was a quartic function, and we were able to solve it through the use of Wolfram alpha, or guess and check, but lead to our later exploration of Polynomial Long Division and Synthetic Division later in the class.
Before we refreshed on these two forms of division, we went over homework, which was composed of problems from section 2.1 and 2.2. since this was our first time looking over chapter two homework together, we refreshed on some basics, like finding the Axis of Symmetry (of quadratics) and and the roots of polynomial functions etc. This will be the material present on next class’ quiz.
Our lesson regarding Synthetic Division and Polynomial Long Division showed us that while both were effective, each had their time and place. Synthetic Division was more simple, but Polynomial Long division could be used regardless of what the polynomial is being divided by. We then saw how the remainder theorem can help one find function values.