Session 18: The first topic from exam period 3 is special triangles. These Khan academy videos use radians rather than degrees. If you are comfortable with radians, then go ahead and watch them. Otherwise consider waiting until after we have had a chance to discuss radians before you view them. There is another Khan Academy set of videos for the same topic but with degrees that I recommend as a supplement or alternative. I’ll admit that I get a little tired of hearing Mr. Khan’s voice, but you may feel differently. Before you watch the video below, you should be comfortable with the idea of ratios, proportions and similar triangles. If not, use the provided links to Khan academy videos or use a web search for the topics.

When watching the videos, note that the easier one to deal with and the one treated first is the 45-45-90 triangle.  Write a summary of all that you know regarding this triangle. What do you know about the legs for this triangle? The most tricky and important special triangle is the 30-60-90 triangle. Write a summary of all that you know regarding this triangle. How do you know which side is the short leg? the long leg? the hypotenuse? The last example is an application. Can you think of another situation with a special right triangle, where you might need to do a calculation? Perhaps construction of a roof with a 30 degree tilt or a building whose interior floor is a hexagon or an octagon?

Session 19: For an introduction to radians and the conversions to and from degrees, see the Khan Academy videos. Here’s Nancy’s video on the same topic. Here is a video on angles in standard position (angles whose initial side is the positive x-axis). Here’s a video on coterminal angles. Given a point, I provided formulas to find the various trig functions for the angles whose terminal side goes through that point. Here’s a video which demonstrates this approach. On Thursday, I will present an alternative approach which is to create right triangles whose legs may be labeled with positive or negative numbers, depending on the quadrant, but whose hypotenuse (radius) is always positive. Once the triangle is labeled, then SOH-CAH-TOA can be used to find the 3 major trig functions. This is my much preferred way as it is visual and this triangle we create will be useful in many other situations, such as in the solution of trig equations.

To create your triangle correctly, you must always go from the point straight towards the x-axis. In other words, the right angle should be either directly above or below the given point. In quadrant II, notice that you have to go straight down. What happens if the point is in quadrant I? How about quadrant III? Quadrant IV? Draw diagrams for each of these scenarios to help you. In any case, the hypotenuse gets labeled by a positive number as it represents the distance from the origin or the radius of the circle centered at the origin which passes through that point.

Session 20: Here are some Khan academy videos to “solve” a right triangle. I like Nancy’s more gentle approach and the drawing on the “window” is a cool effect:

Hopefully you now have a better understanding of what it means to “solve” a right triangle. Write down what you think it means. Also what is the minimal amount and combination of information that needs to be given to “solve” a right triangle? Don’t forget that the right angle has to be provided as part of the given information.

Session 21: We will start the session with review. It has been 2 weeks since we have last had a class and the last topic we covered was done pretty hurriedly. Also, the video above only covers the situation when we are given (besides the right angle) one angle and and one side. To solve a right triangle, there is another situation that occurs namely when you are given 2 sides. They could be 2 legs or they could be one leg and one side. Here is an example that has been worked out nicely:

As you go thru this video. Right down the steps needed to solve. I can get you started.

Given 2 sides of a right triangle:

1. Use the Pythagorean theorem to find the 3rd side.
2. Write a trig function ratio that involves one of the 2 unknown angles and the 2 given sides. For better accuracy, DO NOT USE the 3rd side that you found in step 1.
3. ……….

The first non-review topic for the session is 2 applied notions, namely angle of elevation and angle of depression. For in introduction to the topic, see a video using the Eiffel tower and superman. The following problem has a colorful picture:

One issue for you may be converting degrees and minutes to decimal. Write out a procedure for doing that conversion. For a video on the angle of depression, see this video. Keep in mind that we will use a different technique for solving this kind of problem. Namely instead of using properties of parallel lines, I will create a rectangle and then use a property of rectangles, namely that opposite sides are equal.

Our last topic for the session will be the unit circle and angles associated with the “special” triangles. For the Khan academy treatment, see here. A systematic treatment is

Keep in mind that I will NOT be using this approach exactly. Instead, I will be making a table for the special angles (in the first quadrant) and then using a reference angle approach to get the values in the other quadrants when needed. The quadrantal angles will be treated separately. Here’s an video with an approach closer to what we will use in class:

My only criticism is that the video is using degrees rather than radians. Here is another video that has a similar approach. It goes thru a lot problems very fast:

I would like you to write down a step-by-step procedure that uses her approach. Whether you use her approach or what I do in class will be up to you. The final optional video will close the circle so to speak, as it makes use of the highly divided and labeled unit circle that Nancy created in our first video on the topic.

Session 22: The topic is the graphs of sin and cos (and optionally tangent). Here is the Khan academy approach. There is a lot of material there, but you should at least look at the first video, which is about 9 minutes long. Before we get into our own presentation of the material, I should point that sinusoidal graphs and motion are very important in physics. As an example, here is a graph of the displacement of an object attached you a spring:

The graph shown is an example of what we are mainly studying, namely cosine and sine, which are waves (the tangent is more complicated and will be optional).  Here’s in introduction to our approach:

Note this is generation for sin. I think it visually and systematically shows what’s going on. However, in reality, when we do our graphs in class, we will be plotting fewer points. Namely, we will only plot the quadrantal angle and then from our understanding of what sinusoidal curve is suppose to look like, we then sketch the graph. A video which more closely aligns with what we will do in class is the following presentation of cosine:

Towards the end of Patrick’s video above, he begins to discuss amplitude. Once you have the two basic curves down, then you are ready to flip over the x-axis or change the period or the amplitude. In the next video, Patrick makes all 3 changes.

A more complicated transformation known as a phase shift will be left for MAT 1372 pre-calculus. The following video discusses the phase shift and looks forward towards when you will have to deal with that issue, but treats a problem that just has a period change.

So for your writing assignment, try to create a step-by-step guide to graphing sine or cosine functions with just the 3 changes we are allowing. In particular write down all the formulas you need. Page 146 of your trig text, problems 43 and 47, ask you to go backwards, namely start with a graph and determine its equation. Here’s a gentle introduction. Some more examples appropriate for your level are here. [Note:  webwork problems 1-3 ask you to go backwards as well.]

Finally, if you want to learn a little about the tangent graphs, check out the Khan academy video as well as this alternative.

Session 23: For the Khan academy, here are videos for the fundamental (Pythagorean) identity, including several applications. There are no webwork assignments for the identities themselves. For the sake of those who do not have the textbook, I have a scan of the exercises here. Trig identities is a vast subject and in MAT 1375 precalculus you will learning more of them. Our focus will be on the Pythagorean identity already alluded to, two other closely related identities, which together form the family of Pythagorean identities and some basic ones which you already have been exposed such as the reciprocal relations between sin and csc, cos and sec, tan and cot as well as the fact that tan is the quotient of sin and cos. Let’s start with the Pythagorean Identities. There is a somewhat silly video to help you memorize. The following video has more acceptable-to-a-math teach ring to it:

The bottom line is the best way to deal with Pythagorean Identities is not to memorize them, but to know the first one (the fundamental) and then be able to quickly get the other 2. The presenter in the video shows a quick way to do it by abbreviating sin with s and cos with c. Without referring to the video, see if you can go from the fundamental to the other 2. Write out instructions or justifications for each step. Now that you basically have the tools need, we will go on to proving the results. Considering how dry yet difficult this topic is, Prof Rob Bob has some great energy that can use to focus. He has 3 parts. I have the first one below. You can click on links to the other parts and they will open up in a new window.

Once you watch how he does a problem, recreate it in your notes. Try doing it without looking at the video. Of course if you get stuck, you can always look back. A more advanced exercise would be to try an alternative approach. Did your approach work? In retrospect, which approach will you use on the exam and why? By the way, if you liked Rob Bob’s solutions and energy, here is his playlist for trig.

Session 24: Here is some links to Khan academy videos for sin and cos equations. In particular, look at the first one that exactly matches my solution approach:

In looking for videos, one thing that is evident is that many require knowledge of more advanced trig identities that you will learn in precalculus. However, if you start with the equation AFTER the trig identity is applied, then the problem is something that you will see in our homework, quizzes and exams. The other thing that quickly became evident is that I require a more visual approach than what’s out there. In particular, I will always require a unit circle intersected with a straight line. If the trig function is sine, then it is a vertical line. If the trig function is a cosine, then it is horizontal line. If the function is tangent, then it is a radial line. Brian McLogan’s approach is the closest I found. He has a whole playlist. Below, I have selected two videos which complement the Khan academy video above. When watching these videos, redo them using the method that I provide in class. The first one involves cosine. He solves it the first 2 minutes. The rest of the video relates the solution to the cosine graph and its periodicity. You don’t need it for the solution but it is good to see at least once.

The next video involves tangent:

Reminder that your writing assignment for these last 2 videos is to redo solving these problems using the method that I provide in class.

Session 25: Find one missing piece of triangle using the law of sines or cosines. The webwork assignment is divided in 2. Namely, you have one assignment which requires the law of sines and another assignment requiring the law cosines. Unfortunately, your exams will NOT tell you which method to use, so a good portion of your task is to know which method to use. Start by checking out the Khan academy. There are separate law of sine, law of cosine and oblique triangle pages. Brian McLogan’s videos has a playlist for oblique triangles. First video we will look at is one of videos on the law of sines. The diagram is problematic as I explain later.

Note how long it takes to solve one of these triangles. On exams, we will find just one missing piece (angle or side) rather than solving the whole thing. In terms of patterns, use the law of sines when you are given either AAS or ASA. Given 2 angles, you can easily find the 3rd angle, hence these patterns are equivalent from the solving a triangle point of view. SSA is considered potentially an ambiguous or nonexistent case, but whose solution(s) would be attempted via law of sines as well. In all these cases an angle/opposite side pair are part of the given.

The remaining 2 patterns SSS and SAS use the law of cosines to get one additional piece of information. To completely solve the triangle, a 2nd angle can be found using the law of sines. The 3rd angle could then be found using the fact that the 3 angles of a triangle sum to 180 degrees.

The first part of your writing assignment is to take all the patterns (AAS, etc.) presented in this section and to create a table which organizes how one should proceed when solving a triangle with that given configuration. Again, keep in mind that on an exam, you are just going to be doing the first step. In class we will make a table with that first step.

What follows are videos for the 2 law of cosine cases. First is SSS:

and next is SAS:

In these videos, care is taken to make accurate diagrams, which can then serve as a reality check. Contrast this care with Brian McLogan’s video on the AAS case above. He does not take care in making his diagram accurate. Hence it can not serve as reality check for his answers. Your last writing exercise is to go back and redo his diagram and to make an accurate diagram with the given information. Use the new diagram as reality checks for the answers.