Monday 27 November class

Topics:
• Quick review of the three definitions of the trig functions – we need to use all three from time to time, but the Unit Circle definitions are the most important!

• Important facts that follow from the definitions: sine is an odd function, cosine is an even function, and they satisfy the Pythgorean identity &latex \sin^{2}(x) + \cos^{2}(x) = 1\$.

The main thing when thinking in radians is not to keep translating into degrees (even in your head), but think about what the radian measure angles look like. One full rotation around the circle is $2\pi$; half of that, a straight-line angle, is $\pi$; half of the straight-line angle is a right angle, which is $\frac{pi}{2}$.

To get the other important angles:

We can take half of a right angle, which will be $\frac{\pi}{4}$. This is the acute angle in the isosceles right triangle, because the two acute angles in a right triangle must add up to a right angle. (All 3 angles must add up to a straight line: theorem of geometry!)

We can take a third of a straight line angle, and that will be the angle of the equilateral triangle, which is $\frac{\pi}{3}$ (Again, all three angles must add up to a straight line.) When we cut the equilateral triangle in half to make a right triangle, the smallest angle must be half of $\frac{\pi}{3}$, namely it must be $\frac{\pi}{6}$.

Visualize these angles as you go through this and think about their sizes relative to each other: you don’t have to think of degrees at all! The relative sizes are really very important also and associating them with the radian measure will help you become very fluent in radian language.

• The two important right triangles: in order to avoid thinking in degrees, we will call them by their more proper names: The equilateral right triangle, and the half-equilateral (or hemieq) right triangle. They are shown in this page

Here is a description of how to learn these triangles as I did in class (more or less), good although the drawings are a bit sloppy.

• To learn all the values of the trig functions at the important points on the unit circle: there is a picture here: You can find the blank unit circle to print out and use for practice here: Unit circle for practice

There are patterns in the coordinates of those points, and those may help you to remember them. Here is are some videos from Patrick’s Just Math Tutorials which show how this is done:
A trick to remember values on the unit circle
How to remember all the important points in the unit circle
• Basic graphs of sine, cosine, and tangent. (Was done last time with the sub)
When we consider these functions as functions and look at their graphs, we are always using radians as the input. It is best to think of the input as being a rotation on the unit circle, rather than as an angle. There is a geogebra resource I like to use for the basic graphs: Using the unit circle to generate the graphs.

The trig functions are periodic functions and that is one of the most important things about them. The period is the length along the horizontal axis it takes for the function to start repeating. For sine and cosine, the basic functions have period $2\pi$. The period for tangent is $\pi$.

Sine and cosine functions have two other important parameters: the height from the midline of the wave to the top of the wave is called the amplitude, and the position of the beginning of the first period of the wave is called the phase shift. The phase shift is the amount of horizontal shifting of the basic wave, that is. All three of the parameters have physical significance: for example, for a sound wave, the period is related to the pitch of the sound: the reciprocal of the period is called the frequency. For lightwaves and radiowaves (same thing) the period is called the wavelength. The amplitude has to do with the amount of energy in the wave: for sound waves, large amplitude means a loud sound. The phase shift has to do with what happens when there is more than one wave interacting: when the phases are not the same, the waves can interfere with each other and result in “noise” – microphone feedback squealing is an example of this.

The period, amplitude, and phase can be changed by the same kind of transformations of the basic graphs we have already looked at:
Multiplying the output by a number changes the amplitude
Multiplying the input by a number changes the period
Adding a number to the input changes the phase

Drawing the graph with the important five points: The five-point method

We will do a bit more with this method next time. On the Final Exam you will be asked to sketch the trig graphs and label the five important points.

• New topic: Inverse trig functions. We defined the inverse tangent and the inverse sine functions, following the way they are defined in the textbook (Session 17). We will continue with the inverse cosine function next time, and then do some solving trig equations.

Homework:
• Learn and practice the two important right triangles and the important points on the unit circle – see the videos

• Do the WeBWorK: the second Exponential Functions Applications has been extended to tomorrow, and also start working on the Trig Functions Short: you should do a bit every day or so, not wait until the last minute! (Distributed practice) Remember that when you are asked for the exact values of the trig functions, you should be using the two important triangles and the unit circle picture, NOT your calculator, and also make an effort not to translate into degrees – think in radians.

Don’t forget, if you get stuck on a problem, you can post a question on Piazza. Make sure to give your question a good subject line and tell us the problem itself – we need this information in order to answer your question. And please only put one problem per posted question!

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