Fundamentals of Trigonometry

We begin with a concept from Geometry, and one well-known for millenia. It gives us a method by which we can solve a right triangle, if we have the right information. We will also use this formula to set up the next section.

This section shows us the two ever-important special right triangles: The 45-45-90 and the 30-60-90 triangles. They follow a sort-of-formula that allows us to complete them knowing only one side. We will use these relationships in this section and very, very often in the rest of the course.

We finally begin our first steps into the world of Trigonometry. We meet our first Trig function, the Sine function. We learn that it takes an angle as an input, and outputs a side ratio.

We next meet the cosine function. It is very closely related to the sine function, in both its name and its overall process.

To close out Part I we meet the Tangent function, which we derive from the sine and cosine function. It operates just like the sine and cosine function, but it outputs a different side ratio.

Although not in the textbook, my experience has told me that distinguishing which Trig function to use is a challenge for students. I attempt to show some of the intuition involved in determining whether one should use the sine, cosine, or tangent function in order to complete a right triangle.

You are free to take this test however you like, however, to do this "properly" you should not use any notes, your textbook, and you should not receive help from anyone else. Once you start this test, you must finish it (i.e. you can't do a problem, run an errand, do a few more, then run another errand...). You may use a calculator and you may take as long as you wish.

This test was worth 30 points.

Answer keys are available by request (i.e. email me).

To begin Part II, we take a look at a different way to measure angles. Radians are very natural to use with Trig functions since they are also a ratio. We'll work a little bit with the definition of a radian, but ultimately, the most important thing in this section is to get comfortable with radians, and be able to quickly recognize what pi/6, pi/4, and other fundamental measurements represent.

We introduce perhaps the most important part of Trig: The unit circle. We will build some basic knowledge of the unit circle, and show how it can be used to evaluate Trig functions. To do this, we'll practice finding the coordinates of a point on the unit circle.

We reveal the true usefulness of the unit circle: It allows us to evaluate Trig functions that we don't have memorized, but are awfully similar to ones we already have memorized. This section is challenging, but mastering this content is well worth it.

We combine a host of lessons in this section, so make sure you are comfortable with 2.2, 1.3, 1.4, and 1.5

We consider negative angles in this section. This is neat in and of itself, but it also provides a nice little shortcut for evaluating some Trig functions, particularly those in Quadrant IV.

Up to this point, we've used angles to figure out side lengths. Finally we go in the opposite direction. You'll know a side length ratio, and from that information derive an angle. Thus, we'll learn about the inverse of the Trig functions.

You are free to take this test however you like, however, to do this "properly" you should not use any notes, your textbook, and you should not receive help from anyone else. Once you start this test, you must finish it (i.e. you can't do a problem, run an errand, do a few more, then run another errand...). You may use a calculator and you may take as long as you wish.

This test was worth 30 points.

Answer keys are available by request (i.e. email me).

The goal in this section is to show you identities in the friendliest way possible. This is always a very challenging concept, so we've split it up into two parts, the first of which being very elementary. This section requires intuition and patience, so dig in and be as diligent as possible.

This is a more formal showing of identities, but it's still not at the full scale difficulty you'll encounter in more challenging courses. We spend time distinguishing between equations and identities, which can be very confusing for students first working with identities.

The hope is that these two sections will make you much better prepared for the challenge to come.

Up until now, we've only been using right triangles when working with Trig functions. In the last three sections, we'll work with non-right triangles. The first method we'll use is probably the easiest: The Law of Sines.

The Law of Cosines is able to solve most non-right triangles that the Law of Sines cannot. It's a bit more complicated than the Law of Sines, but it's not that bad if you recognize some of the relationships in the formula.

Perhaps as expected, there is also a Law of Tangents. This Law doesn't give us new functionality, as it is only capable of solving non-right triangles that either the Law of Sines or Law of Cosines can also solve. But the approach may be preferable to some.

This topic is almost never covered or even discussed in any math classes. So while not "required," we hope that you will find it interesting, if not useful.

You are free to take this test however you like, however, to do this "properly" you should not use any notes, your textbook, and you should not receive help from anyone else. Once you start this test, you must finish it (i.e. you can't do a problem, run an errand, do a few more, then run another errand...). You may use a calculator and you may take as long as you wish.

This test was worth 30 points.

Answer keys are available by request (i.e. email me).