The Unit Circle is probably one of the most important topics in all of Trigonometry and is foundational to understanding future concepts in Math Analysis, Calculus and beyond.
The good thing is that it’s fun and easy to learn!
Everything you need to know about the Trig Circle is in the palm of your hand.
In the video below, I’m going to show my simple techniques to quickly Memorize the Radian Measures and all Coordinates for every angle!
- No more trying to calculate all those angles.
- No more getting frustrated when asked to evaluate or memorize each and every coordinate.
Together, we are going to become human calculators, and bring our mathematical genius to life!
What is the Unit Circle?
Well, the Unit Circle, according to RegentsPrep, is a circle with a radius of one unit, centered at the origin.
Why make a circle where the radius is 1, you may ask?
If the radius is a length of 1, then that means that every Reference Triangle that we create has a hypotenuse of 1, which makes it so much easier to compare one angle to another.
Reference Triangle in the First Quadrant of the Unit Circle
But, the Unit Circle is more than just a circle with a radius of 1; it is home to some very special triangles.
Remember, those special right triangles we learned back in Geometry: 30-60-90 triangle and the 45-45-90 triangle? Don’t worry. I’ll remind you of them.
30-60-90 Triangle
45-45-90 Triangle
Well, these special right triangles help us in connecting everything we’ve learned so far about Reference Angles, Reference Triangles, and Trigonometric Functions, and puts them all together in one nice happy circle and allow us to find angles and lengths quickly.
In other words, the Unit Circle is nothing more than a circle with a bunch of Special Right Triangles.
Unit Circle with Special Right Triangles
Now, I agree that may sound scary, but the cool thing about what I’m about to show you is that you don’t have to draw triangles anymore or even have to create ratios to find side lengths.
The Unit Circle
Everything you see in the Unit Circle is created from just three Right Triangles, that we will draw in the first quadrant, and the other 12 angles are found by following a simple pattern! In fact, these three right triangles are going to be determined by counting the fingers on your left hand!
How to Memorize the Unit Circle
Ok, so there are two ways you can do this:
- Use a Unit Circle Chart
- Simply know how to count
If it were me, I’d just want to count and not have to memorize a table, and that’s what I’m going to show you.
The Unit Circle has an easy to follow pattern, and all we have to do is count and look for symmetry. Moreover, everything you need can be found on your Left Hand.
If you place your left hand, palm up, in the first quadrant your fingers mimic the special right triangles that we talked about above: 30-60-90 triangle and the 45-45-90 triangle.
I will show you how to remember each angle, in radian measure, for each of your fingers and also how to find all the other angles quickly by using the phrase:
All Students Take Calculus!
For a quick summary of this technique, you can check out my Unit Circle Worksheets below.
And after you know your Radian Measures, all we have to do is learn an amazing technique called the Left-Hand Trick that is going to enable you to find every coordinate quickly and easily.
Furthermore, this Left-Hand Trick is going to help you not only to memorize the Unit Circle, but it is also going to allow you to evaluate or find all six trig functions!
Additionally, as Khan Academy nicely states, the Unit Circle helps us to define sine, cosine and tangent functions for all real numbers, and these ratios (that we have sitting in the palm of our hand) be used even with circles bigger or smaller than a radius of 1.
Isn’t that awesome?
Yes, indeed!
As you’re watching the video, you’re going to learn how to:
- Draw the Unit Circle.
- Generate every Radian Measure just by counting.
- Use the Left-Hand Trick to find the coordinates of every angle.
- Evaluate all six trigonometric functions for each and every angle on the Unit Circle.
Unit Circle Worksheets
- Blank Unit Circle Worksheet: Practice your skills by identifying the Radian Measure, Degree Measure and Coordinate for each angle.
- How to Memorize the Unit Circle: Summary of how to remember the Radian Measures for each angle.
- Left-Hand Trick: How to find sin cos tan sec csc cot for every angle.
- Unit Circle Chart: Complete Unit Circle with all Degrees, Radian, and Coordinates.
Unit Circle Video
1 hr 38 min
- Intro to Video: Unit Circle
- 00:00:40 – Quick Review of the Six Trig Functions + How to represent them in a Trig Circle
- 00:07:32 – Special Right Triangles & their Importance
- 00:23:51 – Creating the Unit Circle + Left Hand Trick!
- 00:46:37 – Examples #1-7
- 00:55:32 – Examples #8-18
- 01:09:45 – Examples #19-27
- 01:25:35 – Examples #28-36
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