坐标平面中的角：共边角

In trigonometry, so far we have learned about the characteristics of periodic functions and various trig functions like sine, cosine, cosecant, etc. However, now we are going to learn about two new types of angles.

##Reference Angle

Let's take a look at an angle of $330^{\circ}$ in [degrees] or $\frac{11\pi}{6}$ in radians.

To find the reference angle for an angle of $330^{\circ}$, we have to first find what quadrant the terminal side is in. Luckily, there are some nifty rules regarding the quadrant we can use to help determine the amount to subtract from.

If the terminal side is in the first quadrant, you don't have to find a reference angle! Your angle is the reference angle:)

If the terminal side is in the second quadrant, the reference angle is $180^{\circ}$ of $\pi$ minus the angle.

If the terminal side is in the third quadrant, the reference angle is the angle minus $180^{\circ}$ or $\pi$.

If the terminal side is in the fourth quadrant, the reference angle is $360^{\circ}$ or $2\pi$ minus the angle.

In this example, the angle of $330^{\circ}$ is in the fourth quadrant, so know that in order to find the reference angle, we must subtract the angle from $360^{\circ}$ or $2\pi$.

$360^{\circ} - 330^{\circ} = 30^{\circ}$. In terms of radians, $2\pi - \frac{11\pi}{6} = \frac{\pi}{6}$. Therefore the reference angle is $30^{\circ}$ or $\frac{\pi}{6}$.

Looking at this graph, we can see the terminal side and the angle of $330^{\circ}$ labeled, along with the reference angle of $30^{\circ}$. We can see how together, they form a complete circle of a $360^{\circ}$ angle.

Another way of thinking about the reference angle graphically is to find the angle distance between the terminal side and the $x$-axis.

##Coterminal Angle

Coterminal angles are angles that have the same terminal side.

To find the coterminal angle, just add or subtract $360^{\circ}$ or $2\pi$ from the angle.

In the example above, the angle is $30^{\circ}$. If we add $360^{\circ}$, we get $390^{\circ}$, which is a coterminal angle. We can subtract $360^{\circ}$ or add another $360^{\circ}$ as many times as we like and we would always receive a coterminal angle.

(Video) Coterminal Angles

by The Organic Chemistry Tutor

This video by The Organic Chemistry Tutor provides a basic introduction of coterminal angles in degrees & radians.

Summary

Coterminal angles are angles that point to the same terminal angle.

The angle in red is $30 ^{\circ}$, but it is also $390 ^{\circ}$. We can go around the circle multiple times, so $360 + 30 = 390$. $30 ^{\circ}$ is considered the reference angle.

So, anytime two angles differ by $360 ^{\circ}$, you can assume they are coterminal angles. If your angles differ by $2\pi$, you can assume the two angles are coterminal angles.

Coterminal and Reference Angles

A coterminal angle is an angle that shares a terminal side with another angle, but completes more or less rotations around the origin.

In the above image, we see the green angle at $60^{\circ}$. We also see three more coterminal angles drawn, in different colors. The first coterminal angle goes around the origin one additional time, so the coterminal angle is $60^{\circ}+360^{\circ}=420^{\circ}$.

The next coterminal angle wraps around the origin two additional times, compared to the original green angle at $60^{\circ}$. That means that the angle for this coterminal angle is $60^{\circ}+360^{\circ}+360^{\circ}=780^{\circ}$.

This pattern continues for the next coterminal angle. We can calculate the angle, which is $60^{\circ}+360^{\circ}+360^{\circ}+360^{\circ}=1140^{\circ}$.

Now that we understand coterminal angles, we can learn reference angles! You can think of reference angles as the basic of the angle.

The reference angle is always going to be positive, and it will always be an angle between the terminal side and the $x$-axis. It is a measure of the smallest acute angle that a given angle can be.