Wednesday 29 November class

Apologies for the delayed posts while I was getting my computer fixed… I’m working on catching up now

Topics:

• Discussion of exponential functions in applications. The solutions to the quiz are posted here.

• How to find the coordinates of the five important points (the x-intercepts and the maximum and minimum points) in one period of the graph of sine or cosine functions, and sketching the graphs. The notes I used are here:  The five-point method but I am working on making a better version.

• More inverse trig functions: inverse cosine

Please see the notes below for the informal word definitions of the inverse trig functions, which are very useful to say to yourself when you have to find a value of one of them (just as remembering that a logarithm is an exponent is useful).

• Solving a basic tangent equation to find all the solutions.

We will follow a similar procedure for each of the three basic types of trig equations: first we will do an example in some depth in order to invent a method, and then we will apply the method to solve more examples. It is a very good idea to study the first example of each type and make sure that you understand the method before you go on.

For tangent equations, the first example is Ex. 20.1. We get one solution by using the inverse tangent function, and then all the other solutions are found by repeatedly adding or subtracting the period $\pi$ from that first solution.

Definitions of the inverse trig functions (in words):

• To find an inverse to the tangent function, we had to restrict the domain of tangent to the interval $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$, so that will be the range of the inverse tangent function.

Definition: $\tan^{-1}(x)$ or $\arctan(x)$ is the angle (or rotation) in the interval $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ whose tangent is x.

Note: $\tan^{-1}(x)$ is an odd function: $\tan^{-1}(-x) = -\tan^{-1}(x)$

• To find an inverse to the sine function, we had to restrict the domain of sine to the interval $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$, so that will be the range of the inverse sine function.

Definition: $\sin^{-1}(x)$ or $\arcsin(x)$ is the angle (or rotation) in the interval $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ whose sine is x.

Note: $\sin^{-1}(x)$ is an odd function: $\sin^{-1}(-x) = -\sin^{-1}(x)$

• To find an inverse to the cosine function, we had to restrict the domain of cosine to the interval $\left[0,\pi\right]$, so that will be the range of the inverse cosine function.

Definition: $\cos^{-1}(x)$ or $\arccos(x)$ is the angle (or rotation) in the interval $\left[0,\pi\right]$ whose cosine is x.

Note: $\cos{-1}(x)$ is neither even nor odd: but it does satisfy the identity $\cos^{-1}(-x) = \pi -\cos^{-1}(x)$

Homework:

• Review the definitions of the inverse trig functions and how we used them to find exact values. (Session 19.) Note that most of the time we will be working with exact values (not using a calculator) so you need to be able to do this. Review the unit circle if necessary – see the videos from PatrickJMT I linked last time.

There are also useful videos in Khan Academy: introduction to arcsineintroduction to arctangentintroduction to arccosine.

• Study Example 20.1 and make sure that you understand the method it creates for solving tangent equations.

• Do the WeBWorK on inverse trig functions. You are to find exact values in these, so do NOT use your calculator – you will be hurting yourself if you do. Just use your knowledge of the important right triangles and the unit circle picture.

• Find the coordinates of the five important points and sketch the graph for each of these: Exercise 17.6(a-k): I will request volunteers to put these on the board next time, for extra credit (as usual). Note that there is no WeBWorK for this and it will be on the Final Exam, so make sure you practice it!

• There will be a quiz next time: the topic will be finding exact values for the inverse trig functions. You will have to explain how you got your answers, either by drawing the unit circle with its important points or by using the important right triangles. Answers given without explanation will receive no credit!

• Don’t forget that Test 4 is scheduled for next Wednesday. The review self-tests will be in a separate post. This will be a shorter test than usual, so I will add to it a “makeup” problem similar to one that was commonly missed on a previous test. Make sure you look for it. This will give you an opportunity to improve a previous test score.

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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