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The Unit Circle
(click for FREE printable PDF)
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The "Bowtie Angles"™
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The Unit Circle: the "easy way" vs. the "hard way"
Many students, frustrated and confused by the unit circle, choose to instead memorize the unit circle chart to get through it, and in the video(s) below (especially the last couple) I give you lots of tips to help do that. However, there are good reasons to take a more understanding-based approach. For one thing, teachers are good at cooking up questions that the unit circle chart won't work on (I work some of those examples in videos below). Also, since memorization doesn't tend to stick, you'll have to keep re-memorizing the unit circle again for every quiz, test and final.
In my tutoring practice, I've found that the most helpful approach for most students to get through trig in one piece is a "middle way" that combines both memorization and understanding, and I've learned to explain it in a way kids get. By the end of the videos below, you may not be sure if you've memorized the unit circle or you truly understand it, all you'll know is that you're getting the right answers!
Intro to Unit Circle
This first video is mostly about vocab, so we'll explain the terms "Unit Circle", "Reference Angle", "Standard Position", "Quadrants", "Terminal Side", and "Bowtie Angles". We'll do all this in degrees, not radians, because in my experience radians are way easier to deal with if you save them until after you've learned the unit circle in degrees. This unit circle stuff is bad enough on its own!
First Quadrant of Unit Circle
Rather than tackle the unit circle chart all at once, in this video we'll "ease into the pool" by looking at the first quadrant first. I'll show you how to get the first quadrant unit circle values from your special triangles, a key skill that will make the other four quadrants a lot easier to understand and memorize.
"All Students Take Calculus" - the other three quadrants
In this video, we'll take a look at the acronym/pneumonic device that will save your grade whenever you're trying to figure out whether a trig function on the unit circle chart is positive or negative, which is the easiest way to lose points on a quiz or test! A is for All, S is for sine, T is for tangent, and C is for cosine.
How to Calculate Reference Angles
Many students I've tutored have trouble with reference angles, but I think it's mostly because teachers barely teach it, thus making it confusing. In this video I'll teach you a couple of easy-to-remember formulas for calculating reference angles, a skill that will serve you well throughout trig.
Putting It Together: Trig functions of obtuse angles
It's finally time to combine the lessons of the last few videos and find the sine and cosine of angles from the second, third and fourth quadrants! Not exciting, I know, but necessary since it's gonna be on a million tests. From sin135 to cot330 to sec240, by the end of this lesson, you too will be able to derive the unit circle chart using (cosX, sinX)!
More Special Angles - 0, 90, 180, 270
The four "compass points" of the unit circle chart (N, S, E, W) are the angles that have coordinates of 0 and 1 or -1, and they're special because you can't draw a reference angle triangle for them: you've just got to use the unit circle. Sine and cosine aren't too bad for these, but for tangent and the reciprocal functions you've got to choose between zero and undefined, so that's fun. Can't divide by zero!
Sine and Cosine of Negative Angles
This is another topic that I get tons of questions about because teachers seem to give too little attention to it, so I'm giving negative angles their own video. It's easy to say "go around clockwise instead of counterclockwise," but harder in practice, so I work a bunch of examples. For simplicity this video is degrees only, but the concepts are the same for radians.
Co-Terminal Angles (a.k.a. angles bigger than 360)
In this video, we find out how to find sine, cosine, and all the other trig functions of co-terminal angles. The short version is "just take away 360 again and again until it's on the unit circle." (With radians it's the same, we'll just use 2pi instead.)