The Unit Circle

The Unit Circle is perhaps the single most important thing to understand about Trigonometry.  Sure, you can rely on a calculator to get some answers, but sometimes calculators don’t consider all of the possibilities.  If you understand the Unit Circle, you will.

The Unit Circle will help you understand how to find the sine, cosine, or tangent, of an angle greater than 90 degrees, help you find tangent of things like 90 degrees (which seems impossible because tangent is opposite over adjacent), and is a great reference for finding common Trigonometric values.

The Unit Circle is constructed from a pair of special right triangles.  This is why we consider knowledge of those triangles analogous to arithmetic.  It all starts with the 30 – 60 – 90 and 45 – 45 – 90 right triangles!

Read through the notes, taking notes yourself.  Download the PowerPoint and play it.  Give yourself the patience required to understand the information.  Watch the video, then try the practice problems.  

If you have questions or confusion about the Unit Circle, please send an email with your question.  Send emails to:  thebeardedmathman@gmail.com.

The Unit Circle

The picture above is called the Unit Circle.  It is a diagram of trigonometric values that can be used as a reference for basic trigonometric function values.  In this section we will break it down, examine its parts so that you can not only understand it, but can recall the values without memorization. 

Let’s start with something that might be completely new to you.  There are angles written in degrees, with the degree symbol (°), but there is another unit of measure for angles called radians.  Radians will be discussed in detail later, but for now just understand that 360° = 2π radians.

Let’s break down what the Unit Circle is, how it is constructed.

First:  The Unit Circle is a circle with a radius of 1 (unit), whose center is at the origin of a coordinate plane (x, y). 

Second:  Because the radius is one, where the edges of the circle cross the x and y axes, we have the following coordinates.

Third:  Angles on the unit circle are rotations around the origin, starting at the positive x – axis, in a counter-clockwise direction.  (Angles are measures of rotation around a point.)

Fourth: If we rotate either 30°, 45°, 60°, or a multiple of those, we can create a right triangle whose properties we know well (from the last section).  We can use this information to find the coordinate at the edge of the circle after a rotation of θ°.

For reference, the properties of the isosceles right triangle is in the diagram below.  In short, whatever the hypotenuse is, the legs are that times the “square root of two over two.”

Let’s take a look at another angle, 150°.  You might be wondering how we can use the triangles we know to find this.  We use what’s called a reference angle.  A reference angle is the positive acute angle that can represent any other angle of measure. 

You know, sometimes picture good, word bad.

We will use what we know about a 30°, 60°, 90° triangle to find the coordinate where the ending side of this triangle crosses the circle’s edge.

The coordinate at 150° of rotation, starting from (1, 0), and rotating counter clockwise is

The reason the x – coordinate is negative is because this is in the 2nd quadrant.  The side lengths are known because of the properties we explored about a 30°, 60°, 90° triangle.

Let’s see one last example, this time in quadrant III. 

We need to make a reference angle with one of the axes.  We have two choices.  We can either make the angle going towards the y – axis, or going towards the x – axis.  We already did an example using a 30°, so let’s make a 60° reference angle here.  Don’t forget that the radius is 1.

Summary:

  1. The unit circle has a radius of 1 and a center at (0, 0).
  2. The radius of the circle is the hypotenuse of a right triangle.
  3. The rotation starts at (1, 0), and rotates counter clockwise.
  4. Reference angles, which are acute, are created between the ending point and an axis. We use these acute angles to find the side lengths of a right triangle, whose hypotenuse is the radius of the circle.

 

What does all of this have to do with trigonometry?

The sine of 30 degrees is “opposite over hypotenuse.”  That reduces to 1/2, regardless of the size of the hypotenuse. 

The cosine of 30 degrees is “adjacent over hypotenuse.” That reduces to “root three over two”, regardless of the size of the hypotenuse.   Do you see how the Unit Circle shows you this information?  The x – coordinate is cosine, and the y – coordinate is sine.

Do you see how you can use the Unit Circle to

find the sine of 90°?  It is 1.  What about the cosine of 0°?  

What about tangent of 30°?  Since tangent is opposite over adjacent, this is also sine divided by cosine!

Let’s see if that is always true.

 

Last thing you need to know, before we move on. Other than ½, 1, and 0, the values on the unit circle are all irrational.  Most inexpensive calculators use approximations for irrational numbers.  You’ll be using a calculator to verify your practice problem answers, and also reinforcing the connection between the Unit Circle and trigonometric values.  Here are the approximations to 3 decimal places.

Note:  There are short-cuts that can be used to remember the Unit Circle, but they’re not appropriate to be used yet. 

The Unit Circle explains a lot of confusing things about Trigonometry for us.  Before we get into that, do you see how the Table of Trigonometric Values is created from the Unit Circle?

The Unit Circle also helps us see:

  • Why the tangent of 90º and 270º are undefined
  • How we can use the ratio of sides of a triangle to find the sine of 210º
  • Why some trig values are negative

Many modern treatments of Trigonometry avoid dealing with the Unit Circle because the use of calculators is prevalent.  But, by understanding the Unit Circle you gain insight into some of the issues highlighted above and also practice the fundamentals of Trigonometry and some procedures involved with the subject.

One last question:  Use the Unit Circle to find the angles whose sine is 1/2.  There are more than one.  In trig this would be sin-1(1/2).  Use a calculator to evaluate this.  Many calculators give you only one answer.

Here you can download a lesson that helps you understand and then construct a Unit Circle.  To download the PowerPoint, please click here.

To download a printable copy of the Unit Circle, please click here.