# Trigonometry: The Unit Circle [in Plain English]

## The Idea:

The **unit circle** lets us visualize the coordinates of a circle on a graph. Of course, there are lots more things the unit circle is used for, but we will get into them later. The important thing to realize is that the unit circle is *just a picture of a circle* with a radius of one! This helps us to see the connection between the **Pythagorean Theorem** (A^{2} + B ^{2} = C^{2 }) and **sines**, **cosines**, and **tangent**.

In this article, we will learn how to

- Construct a unit circle
- Find the sine or cosine of any angle
- Use angles in degrees and radians

## The Unit Circle

## Constructing a Unit Circle

For now, we will only focus on the** first quadrant**, which is the upper-right part of the graph. Notice that there is a line going up at an angle, from the center of the circle (the **origin**) to the edge of a circle. It is going up at 30^{o}, touching the circle at the point (^{√3}/_{2}, ^{1}/_{2}). These two numbers are the cosine(30) and the sine(30), respectively. *So how does sin(30) = 1/2? *

Let's draw a picture.

## Sin(30): In a Picture

## Let's Break it Down

Here are some important things to remember:

**Sine**= the ratio of the opposite side of a triangle to its**hypotenuse**, or longest side**Cosine**= the ratio of the adjacent side of a triangle to its hypotenuse- When we say opposite or adjacent, we mean
*with respect to the angle we're measuring*

When we draw a line from the origin to a point on the circle, it creates a little triangle with the side lengths given by the coordinates of where it touches. Since the hypotenuse is always 1 on the unit circle, the value of the sine and cosine are simply whatever the opposite and adjacent side lengths are. That's it!

**Note:** If we choose the other angle, 60^{0}, to be what we find the sine of, the value of the sine and cosine would just be reversed.

**Also Note:** No matter what point we choose on the circle, the sum of its squares will always equal 1. This is where the trig identity sin^{2}(x) + cos^{2}(x) = 1 comes from: an alternate form of the Pythagorean Theorem. Test the answers we found above to confirm the theorem!

Now that we know that sin(x) = opposite/hypotenuse and cos(x) = adjacent/hypotenuse (x represents any angle our line makes with the X-axis), we can find all the points where our line touches the circle. All we need to know is the angle the line is making with the X-axis.

Notice that the values of cosine and sine switched from our previous example! In fact, the value of sine and cosine alternate between just a few values for the common angles used on the unit circle. Here is the complete circle:

## Why can I have a positive cos(x) with a negative angle?

## Using Radians

At some point, you may encounter a strange looking unit called a **radian** that's used to measure an angle, usually expressed as some form of π. You might need to convert from one unit to another, and take the sine or cosine of a radian measurement. It's actually quite simple!

**Steps:**

- First, note that 2π = 360
^{o}. This means that for every rotation around the circle, we go 2π, or about 6.28, radians. (We try to keep all our radians in terms of π). - To convert degrees to radians, multiply by 2π/360.
- To convert radians to degrees, multiply by 360/2π.

This works because the ratio of radians to degrees remains the same, so we can just use unit math with fractions to get the degrees or radians to drop out - leaving us with our desired unit! This approach of cancelling units works for many, many types of problems from physics to chemistry, and is well worth mastering.