Use the blank unit circle worksheet to test yourself and keep the filled unit circle handy for a reference.

Memorizing these common values makes for a valuable tool for any mathematician. From trig to geometry and calculus, these values will come in handy many times over.

## Blank Unit Circle Worksheet

## Unit Circle Reference

Look for patterns in the values to help memorize them.

### Why Does the Circle have a Unit?

What makes it special is having a radius of 1, which is a *unit* in our number system. The circle is centered on the origin, *x*=0, *y*=0, or coordinates (0,0).

These simple settings give lengths and sin/cos/tan values that are easy to work with compared to neverending decimal numbers, and are very useful, as 30°, 45°, and 60° angles show up a lot in the world!

## Pythagoras

Just like many topics, we’re back to Pythagoras.

The Pythagorean Theorem says that adding the square of either side is equal to the square of the hypotenuse for a right-angle triangle.

For the unit circle, the hypotenuse is always 1, so the square of the hypotenuse is also 1.

*x*

^{2}+

*y*

^{2}= 1

Which turns out to be the equation for the unit circle!

You can take this one step further – from the parametric equation *x*=cos*θ* and *y*=sin*θ*, so the equation for the unit circle is also

*θ*)

^{2}+ (sin

*θ*)

^{2}= 1

This is one of the useful trigonometric identities.

## Useful Values to Remember

Here is a table of useful values to remember.

Memorization can be boring work, but here is a trick to help you.

The **sin** values for 30°, 45°, and 60° are:

The **cos** values for 30°, 45°, and 60° are:

Finally, tan=sin/cos so all you *really* need to remember are the sin values.

## Summary

The unit circle has a radius of 1 and is centered on the origin, (0,0).

The values of sin, cos, and tan for 30°, 45°, and 60° are given by radicals – easier to work with than unwieldy decimal numbers.

The equation for the unit circle is:

*x*

^{2}+

*x*

^{2}= 1OR

(cos *θ*)^{2} + (sin *θ*)^{2} = 1

We hope the worksheet and reference sheet come in handy for you or your learners.

How long did it take you to fill in your blank unit circle? Let us know in the comments, or show off by typing out the sin, cos, and tan values for 30°, 45°, and 60° without looking at the reference sheet!

DaveJesse,

Could you please change the equation for the unit circle to x^2+y^2=1. On my computer it was x^2+x^2=1. Thank you. By the way, your website is excellent.

Dave Loy

Jesse WoodsHi Dave,

It’s been updated, good spot! Thank you for your kind words, it’s nice to hear that my efforts are appreciated.

I hope you stay well!

Jesse

KateThis is so helpful! Thank you!

Jesse WoodsHi, Kate.

You’re so welcome! It’s always a pleasure to hear that these resources are appreciated.

I hope you are well.

Jesse