UPDATED
First, two important notes:
* For the WeBWorK on trig equations (which has been reopened), in problems 3 and 4 there is an issue which David brought to my attention yesterday. Notice that both of these problems ask for the principal solution to a trig equation, but unlike the equations we have solved in class, they do not specify that the solution should be in $[0,2\pi)$. In that case, if the equation is of the form $\sin{x} =$ negative number or $\tan{x} =$ negative number, the principal solution is going to be the negative angle which is coterminal with the angle in our Unit Circle, which is easy to find because it’s just the negative of the reference angle!
This applies ONLY to sine and tangent (not cosine) and ONLY when we do not specify that the solutions should be in $[0,2\pi)$.
* On the matter of “Do not Google”, what to do?
First use the “Ask for help” button in WeBWorK!
Next, realize that I have already got some reliable resources linked on the resources page.
If you do go ahead and google despite all this, PLEASE send me a link to what you found so I can check it out. Remember that I am your Main Resource and I am always willing to work with you! (Plus, you might find a great resource I don’t already know about!)
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These notes and handouts are (mainly) on the topics:
- Unit Circle definition of trig functions,
- Basic graphs of sine and cosine functions
- Solving trig equations using the Unit Circle
- Trig identities and proving identities: the method taught in class is specially designed to eliminate one of the big hurdles that students encounter in this topic and make it much simpler! But, you will need to practice working with rational expressions – adding (changing to a common denominator), multiplying, and reducing
Here are the handouts and other materials:
Here are two nice videos from Patrick’s Just Math Tutorials showing how to remember the important points in the Unit Circle easily: they correspond closely to what we did in class.
MAT1275-WhatsMyAngle-slideshow (Shows how we got from the special angles in the first quadrant to each of the other quadrants by reflecting the reference triangle)
MAT1275SolvingTrigEquationsActivitiesHandout
Proving trig identities made easier (handed out in class, but this version has all the colors to help make clear what we are doing!)
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Notes on the basic graphs of sine and cosine functions:
We used the unit circle to see what the graph of $y = \sin{(x)}$ looks like. Here is a Geogebra animation which shows both the sine and cosine graphs being generated by rotating around the unit circle. Remember that here x is the angle (in radians) and the coordinates of the point on the unit circle are cosine and sine.
We then looked at what happened to the graph of $y = \sin{(x)}$ when we made certain types of transformations. I suggest you graph $y = \sin{(x)}$ in Desmos and then compare to the graphs of each of the following:
$y = 2\sin{(x)}$
$y = \frac{1}{2}\sin{(x)}$
$y = -3\sin{(x)}$
$y = \sin{(2x)}$
$y = \sin{(\frac{1}{2}x)}$
$y = 3\sin{(\frac{1}{4}x)}$
We deduced from these experiments that multiplying the sine function by a number changes the amplitude of the graph, and multiplying the input angle x by a number changes the period. Specifically, $y = A\sin{(Bx)}$ has amplitude $|A|$ and period $\frac{2\pi}{B}$ (if B is positive, which in real life applications it always is).
So considering for example $y = -5\sin{(\frac{2}{3}x)}$,
here $A = -5$ and $B = \frac{2}{3}$
so the amplitude is $|-5| = 5$ (amplitude is ALWAYS positive!)
and the period is $\frac{2\pi}{\left(\frac{2}{3}\right)} = 2\pi\left(\frac{3}{2}\right) = 3\pi$
