The unit circle is a fundamental concept in math, and will be especially helpful in non-calculator papers involving solving trigonometric functions. Defined as a circle with a radius of , it is centered around the origin in the Cartesian plane. From the equation of a circle - - we can extract the starting point of a useful identity that will be discussed later: .

For now, I don't have tikz rendering set up, however you should draw a circle with the aforementioned qualities now. Then, draw some diagonal line, pointing upwards diagonally, from the origin until it touches the circumference of the circle. Note that the line touches a point on the circle. Now, try and find the and components of that point of intersection given the angle which you should also draw that starts from the horizontal positive -axis going counterclockwise to that line.

You should now realize that the component is and the component is . We can therefore re-write the coordinate of that point as and start looking at how we can solve trigonometric values without needing a calculator!

If you think about it, is perpendicular to the -axis, hence why its value is 0 (that is, , because there's no movement in the direction). The same can be said for , but as it's the -value, it's going to be the largest, hence why . For other simple values, you'll either have to learn them from the table below, or through drawing triangles (again, once tikz is set up, I'll demonstrate this visually).

You'll only have to memorize values up to , as past that you can just use the unit circle to determine the values.