Right Angle Trig on the Unit Circle

Unit Circle: What is it?

The Unit Circle is sort of what it sounds like. It is a circle whose radius is exactly one unit long. What unit you ask? It doesn't matter, one inch, one kilometer or one nanometer. Just so long as its one. Most of the time math folk don't even give it a unit… its just 1. Kind of like how I didn't put an apostrophe in "its" - we still all know what I meant even if its not 100% awesome.

The unit circle is also centered on the origin. This gives it the equation $x^2+y^2=1$ which while useful is not something SL students are expected to know. The unit circle looks something like this:


The unit circle can be your friend and help you solve lots of problems or it can be a royal pain… I suggest making friends with it.

I See Triangles

When talking about the unit circle we frequently, rather always, are drawing a radius from the origin to some point on the circle. The point where that radius meets the circle has $(x,y)$ coordinates. These coordinates are most often what we are interested in. On the diagram below is a radius intersecting the circle at a point. Lets pretend we want to find the coordinate points of the intersection. This could be done knowing the equation of the radius (linear function) and the equation of the circle and using your calculator… Or we can use some Right Angle Trig (RAT).

We can create a right triangle as shown below. Notice that the length of the adjacent side is the x-coordinate and the length of the opposite side is the y-coordinate.


From this we can write a couple statements:

\begin{align} sin 30^\circ = \frac{opp}{hyp}=\frac{y}{r} \end{align}
\begin{align} cos 30^\circ = \frac{adj}{hyp}=\frac{x}{r} \end{align}

Because the radius (r) of the circle is 1 the statements both simplify. Which leads us to the next section.

Defining Sine, Cosine and Tangent on the Unit Circle

Using mathematical statements above we can define the sine and cosine functions on the unit circle as:

\begin{align} sin \theta = y \end{align}
\begin{align} cos \theta = x \end{align}

Be careful. These definitions are only valid when talking about the unit circle. If the radius is not 1 then the statements can not be simplified as much.

We can go one step further and define the tangent function on the unit circle.

\begin{align} tan \theta = \frac{opp}{adj}=\frac{y}{x} \end{align}

From this last statement its also pretty easy to see or say that:

\begin{align} tan \theta = \frac{sin \theta}{cos \theta} \end{align}

Which the IB refers to as the "Trigonometric Identity" and is in your data booklet.

What You Need To Memorize

If you are taking your exams in 2014 or later you need to memorize the following:

Degrees 0 30 45 60 90
Radians 0 $\frac{\pi}{6}$ $\frac{\pi}{4}$ $\frac{\pi}{3}$ $\frac{\pi}{2}$
$sin(\theta)$ 0 $\frac{1}{2}$ $\frac{\sqrt{2}}{2}$ $\frac{\sqrt{3}}{2}$ 1
$cos(\theta)$ 1 $\frac{\sqrt{3}}{2}$ $\frac{\sqrt{2}}{2}$ $\frac{1}{2}$ 0
$tan(\theta)$ 0 $\frac{1}{\sqrt{3}}$ 1 $\sqrt{3}$ undefined

The truth is you must memorize the entire unit circle. BUT! There are patterns. If you can memorize the values above then remember the patterns it can greatly reduce what you need to memorize. The entire unit circle with sine and cosine values is shown below. Tangent values can be found by remembering that $tan \theta =\frac{sin \theta}{cos \theta}$.


Tips to Memorize the Unit Circle

I hate. I. HATE. Math gimmicks. They cheapen a subject that most students already hate… But since the IB wants you to memorize stuff here's a good gimmick (trick) to memorize or remember the unit circle.

Another good video suggested by a reader. Pretty clever and simple way to "calculate" exact values from 0 to 90 degrees for sine, cosine and tangent. Actually makes use of some cool patterns…

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