The unit circle is the golden key to actually understanding trigonometry. Like many ideas in math, it is simplicity that makes it beautiful.
But, before we go off on a tangent – get the chart you came here for.
The unit circle is a circle centered on the origin with a unit radius, 1.
Sine, Cosine, Tangent
The measurements of sin, cos, and tan become clear when you see them on a graph. Take a moment to soak in what they mean.
Sin, Cos, Tan at 0°
So, what are the sin, cos, and tan of an angle of 0°?
Sin, Cos, Tan at 90°
And what about 90°?
Special Angles: 30°, 45°, and 60°
A unit circle shows trig functions clearly because the radius is 1. The hypotenuse doesn’t change the value of sin, cos and tan.
The angles 30°, 45°, and 60° have special properties for sin, cos and tan.
A little time spent memorizing them now will save you a LOT of time, doing your working out in the future.
Memorizing sounds like a pain, but don’t worry, there are some tricks to help. Let’s start with the values for sin.
These are the only values you need to memorize. Can you see why?
Values for cos are the sin-values in reverse!
The final trick is this:
This can be easily remembered by thinking of a stop sign, and cos lettuce. See where this is going?
Tan also leads to a nice pattern, although it doesn’t include 0° and 90° like sin and cos do.
Put these all together and you get the table of special trigonometric values, or the unit circle table:
Alternative Method to Rememeber
You may prefer this one, because you can figure it out yourself, even in the middle of an exam!
First, you need the tool – a special 30-60-90 triangle. There are three steps:
- Draw an equilateral triangle with side length 2
- Split it down the middle
- Use Pythagoras’ Theorem to find the new side’s length
Then, use SOH CAH TOA on the triangle. Remember that each internal angle of an equilateral triangle is 60°, so the halved angle is 30°.
The final piece of the puzzle is which sign to use around the circle, + or -.
The Four Quadrants of the Unit Circle
A quadrant is a quarter of a circle, so there are 4 of them.
The value of sin, cos, and tan stay the same in each quadrant, but the sign changes depending on which quadrant the angle is in.
Okay, so you have everything needed to build a complete picture.
Unit Circle Chart
Take everything you’ve seen so far:
- The values for the special angles, 30°, 45°, and 60°
- cos = x
- sin = y
- The positive and negative values for each quadrant
And put them all together. It leads to this very handy chart. Click/Tap on the image to bring up a printable PDF.
There it is! The values that include pi, π, are called radians. They’re the next step on the road to mastery of the unit circle.
Are you looking for a blank unit circle worksheet?