Trig+Identities+and+Exploration+48+10-3

In today's class, we finished up half-angle formulas for tan(x) and looked at the results of multiplying or adding sinusoidal functions.


 * __Half-Angle Formulas__**

Half-angle formulas express the trigonometric function of an angle //x/2// in terms of angle //x//. The half-angle formulas for sin(x), cos(x), and tan(x) are as follow. These are all on the identity handout provided in class:








 * Note: Mrs. Tyson pointed out that to quickly work out identities, plus/minus signs are unnecessary, but when working through or writing out a proof, plus/minus signs are necessary for justification.**

The following are the steps we took to derive one of the half-angle formulas from tan (x/2):

Using this, we can obtain the next half-angle formula for tan(x/2) through the following steps:



In the above equation, the plus/minus signs can be removed between the last two steps because sin(x) and tan(x/2) will always have matching signs. The absolute value bars can also be removed from both the numerator and denominator because 1-cosx is always greater than or equal to 0, and sin(x) is positive in [0, pi]

We took this further by multiplying the above result by the conjugate of its numerator to get:

Next, we worked on a handout Ms. Tyson provided.

__**Exploration 48: Sums/Products of Sinusoids with Unequal Periods**__

This is the graph of y=9sinx, y=cos9x, and the **product** of the two (see key). For the above graph, we realized that because the yellow function has a constantly changing amplitude (for this function, amplitude is a function), we know that the function is 9sinx * cos9x and not 9sinx + cos9x.

This is the graph of y=9sinx, y=cos9x, and the **sum** of the two (see key). For this graph, we realized that the graph of y=9sinx + cos9x oscillates around 9sinx, and because of that we know that the function is 9sinx + cos9x and not 9sinx * cos9x.

__**CONCLUSIONS:**__
 * //Adding// sinusoids with different periods produces //a different axis of oscillation.//**
 * //Multiplying// sinusoids with different periods produces //a different amplitude.//**

(all graphs from Wolfram Alpha)