What is the Unit Circle in Trigonometry?

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Trigonometry can feel like quite an abstract subject. Arcane terms like “sin” and “cos” just don’t seem to correspond to anything in reality, and it’s hard to get a grasp on them as concepts. The unit circle helps substantially with this, offering a straightforward explanation of what the numbers you get are when you take the sine, cosine or tangent of an angle. For any students of science or math, understanding the unit circle can really cement your understanding of trigonometry and how to use the functions.

TL;DR (Too Long; Didn't Read)

A unit circle has a radius of one. Imagine an ​xy​ coordinate system starting at the center of this circle. The point angles are measured from is where ​x​ = 1 and ​y​ = 0, on the right hand side of the circle. Angles increase as you move counter-clockwise.

Using this framework, and ​y​ for the ​y​-coordinate and ​x​ for the ​x​-coordinate of the point on the circle:

sin ​θ​ = ​y

cos ​θ​ = ​x

And consequently:

tan ​θ​ = ​y​ / ​x

What Is the Unit Circle?

A “unit” circle has a radius of 1. In other words, the distance from the center of the circle to any part of the edge is always 1. The unit of measurement doesn’t really matter, because the most important thing about the unit circle is that it makes many equations and calculations much simpler.

It also serves as a useful basis for looking at the definitions of angles. Imagine that the center of the circle sits at the center of a coordinate system with an ​x​-axis running horizontal and a ​y​-axis running vertically. The circle crosses the ​x​-axis at ​x​ = 1, ​y​ = 0. Scientists and mathematicians define the angle from that point moving in a counter-clockwise direction. So the point ​x​ =1, ​y​ = 0 on the circle is at an angle of 0°.

The Definitions of Sin and Cos With the Unit Circle

The ordinary definitions of sin, cos and tan given to students relate to triangles. They state:

\sin θ = \frac{\text{opposite}}{\text{hypotenuse}} \\ \,\\ \cos θ = \frac{\text{adjacent}}{\text{hypotenuse}} \\ \,\\ \tan θ = \frac{\sin θ}{\cos θ}

The “opposite” refers to the length of the side of the triangle opposite the angle, “adjacent” refers to the length of the side next to the angle and “hypotenuse” refers to the length of the diagonal side of the triangle.

Imagine creating a triangle so that the hypotenuse was always the radius of the unit circle, with one corner at the edge of the circle and one at its center. This means that hypotenuse = 1 in the equations above, so the first two become:

\sin θ = \frac{\text{opposite}}{1} = \text{opposite}\\ \,\\ \cos θ = \frac{\text{adjacent}}{1} = \text{adjacent} \\

If you make the angle in question the one at the center of the circle, the opposite is just the ​y​-coordinate and the adjacent is just the ​x​-coordinate of the point on the circle that touches the triangle. In other words, sin returns the ​y​-coordinate on the unit circle (using coordinates that start at the center) for a given angle and cos returns the ​x​-coordinate. This is why cos (0) = 1 and sin (0) = 0, because at this point those are the coordinates. Likewise, cos (90) = 0 and sin (90) = 1, because this is the point with ​x​ = 0 and ​y​ = 1. In equation form:

\sin θ = y \\ \cos θ = x

Negative angles are also easy to understand on the basis of this. The negative angles (measured clockwise from the starting point) have the same ​x​ coordinate as the corresponding positive angle, so:

\cos -θ = \cos θ

However, the ​y​-coordinate switches, which means that

\sin -θ = -\sin θ

The Definition of Tan With the Unit Circle

The definition of tan given above is:

\tan θ = \frac{\sin θ}{\cos θ}

But with the unit circle definitions of sin and cos, you can see this is equivalent to:

\tan θ = \frac{\text{opposite}}{\text{adjacent}}

Or, thinking in terms of coordinates:

\tan θ = \frac{y}{x}

This explains why tan is undefined for 90° or −270° and 270° or −90° (where ​x​ = 0), because you can’t divide by zero.

Graphing Trigonometric Functions

Graphing sin or cos becomes easier when you think of the unit circle. The ​x​-coordinate varies smoothly as you move around the circle, starting at 1 and decreasing to a minimum of −1 at 180°, and then increasing in the same way. The sin function does the same thing, but it increases to a maximum value of 1 at 90° first, before following the same pattern. The two functions are said to be 90° out of “phase” with each other.

Graphing tan requires dividing ​y​ by ​x​, and so is more complicated to graph, and also has points where it is undefined.

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