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# 1.18: Reference Angles and Angles in the Unit Circle

Difficulty Level: At Grade Created by: CK-12
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Practice Reference Angles and Angles in the Unit Circle
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When you walk into math class one day, your teacher has a surprise for the class. You're going to play series of games related to the things you've been learning about in class. For the first game, your teacher hands each group of students a spinner with an "x" and "y" axis marked. The game is to see how many angles you identify correctly. However, in this game, you are supposed to give what is called the "reference angle". You spin your spinner three times. Each picture below shows one of the spins:

Can you correctly identify the reference angles for these pictures?

At the end of this Concept, you'll know what reference angles are and be able to identify them in the pictures above.

### Guidance

Consider the angle 150\begin{align*}150^\circ\end{align*}. If we graph this angle in standard position, we see that the terminal side of this angle is a reflection of the terminal side of 30\begin{align*}30^\circ\end{align*}, across the y\begin{align*}y-\end{align*}axis.

Notice that 150\begin{align*}150^\circ\end{align*} makes a 30\begin{align*}30^\circ\end{align*} angle with the negative x\begin{align*}x-\end{align*}axis. Therefore we say that 30\begin{align*}30^\circ\end{align*} is the reference angle for 150\begin{align*}150^\circ\end{align*}. Formally, the reference angle of an angle in standard position is the angle formed with the closest portion of the x\begin{align*}x-\end{align*}axis. Notice that 30\begin{align*}30^\circ\end{align*} is the reference angle for many angles. For example, it is the reference angle for 210\begin{align*}210^\circ\end{align*} and for 30\begin{align*}-30^\circ\end{align*}.

In general, identifying the reference angle for an angle will help you determine the values of the trig functions of the angle.

#### Example A

Graph each angle and identify its reference angle.

a. 140\begin{align*}140^\circ\end{align*}

b. 240\begin{align*}240^\circ\end{align*}

c. 380\begin{align*}380^\circ\end{align*}

Solution:

a. 140\begin{align*}140^\circ\end{align*} makes a 40\begin{align*}40^\circ\end{align*} angle with the negative x\begin{align*}x-\end{align*}axis. Therefore the reference angle is 40\begin{align*}40^\circ\end{align*}.

b. 240\begin{align*}240^\circ\end{align*} makes a 60\begin{align*}60^\circ\end{align*} with the negative x\begin{align*}x-\end{align*}axis. Therefore the reference angle is 60\begin{align*}60^\circ\end{align*}.

c. 380\begin{align*}380^\circ\end{align*} is a full rotation of 360\begin{align*}360^\circ\end{align*}, plus an additional 20\begin{align*}20^\circ\end{align*}. So this angle is co-terminal with 20\begin{align*}20^\circ\end{align*}, and 20\begin{align*}20^\circ\end{align*} is its reference angle.

#### Example B

Find the ordered pair for 240\begin{align*}240^\circ\end{align*} and use it to find the value of sin240\begin{align*}\sin 240^\circ\end{align*}.

Solution: sin240=32\begin{align*}\sin 240^\circ = \frac{-\sqrt{3}}{2}\end{align*}

As we found in Example A, the reference angle for 240\begin{align*}240^\circ\end{align*} is 60\begin{align*}60^\circ\end{align*}. The figure below shows 60\begin{align*}60^\circ\end{align*} and the three other angles in the unit circle that have 60\begin{align*}60^\circ\end{align*} as a reference angle.

The terminal side of the angle 240\begin{align*}240^\circ\end{align*} represents a reflection of the terminal side of 60\begin{align*}60^\circ\end{align*} over both axes. So the coordinates of the point are (12,32)\begin{align*}\left( -\frac{1}{2}, -\frac{\sqrt{3}}{2} \right )\end{align*}. The y\begin{align*}y-\end{align*}coordinate is the sine value, so sin240=32\begin{align*}\sin 240^\circ = -\frac{\sqrt{3}}{2}\end{align*}.

Just as the figure above shows 60\begin{align*}60^\circ\end{align*} and three related angles, we can make similar graphs for 30\begin{align*}30^\circ\end{align*} and 45\begin{align*}45^\circ\end{align*}.

Knowing these ordered pairs will help you find the value of any of the trig functions for these angles.

#### Example C

Find the value of cot300\begin{align*}\cot 300^\circ\end{align*}

Solution: cot300=13\begin{align*}\cot 300^\circ = -\frac{1}{\sqrt{3}}\end{align*}

Using the graph above, you will find that the ordered pair is (12,32)\begin{align*}\left ( \frac{1}{2}, -\frac{\sqrt{3}}{2} \right )\end{align*}. Therefore the cotangent value is cot300=xy=1232=12×23=13\begin{align*}\cot 300^\circ = \frac{x}{y} = \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = \frac{1}{2} \times -\frac{2}{\sqrt{3}} = -\frac{1}{\sqrt{3}}\end{align*}

We can also use the concept of a reference angle and the ordered pairs we have identified to determine the values of the trig functions for other angles.

### Vocabulary

Reference Angle: A reference angle is the angle formed between the terminal side of an angle and the closest of either the positive or negative 'x' axis.

### Guided Practice

1. Graph 210\begin{align*}210^\circ\end{align*} and identify its reference angle.

2. Graph 315\begin{align*}315^\circ\end{align*} and identify its reference angle.

3. Find the ordered pair for 150\begin{align*}150^\circ\end{align*} and use it to find the value of cos 150\begin{align*}150^\circ\end{align*}.

Solutions:

1. The graph of 210\begin{align*}210^\circ\end{align*} looks like this:

and since the angle makes a 30\begin{align*}30^\circ\end{align*} angle with the negative "x" axis, the reference angle is 30\begin{align*}30^\circ\end{align*}.

2. The graph of 315\begin{align*}315^\circ\end{align*} looks like this:

and since the angle makes a 45\begin{align*}45^\circ\end{align*} angle with the positive "x" axis, the reference angle is 45\begin{align*}45^\circ\end{align*}.

3. Since the reference angle is 30\begin{align*}30^\circ\end{align*}, we know that the coordinates for the point on the unit circle are (32,12)\begin{align*}\left( -\frac{\sqrt{3}}{2},\frac{1}{2} \right) \end{align*}. This is the same as the value for 30\begin{align*}30^\circ\end{align*}, except the "x" coordinate is negative instead of positive. Knowing this,

cos150=adjacenthypotenuse=321=32\begin{align*} \cos 150^\circ = \frac{adjacent}{hypotenuse} = \frac{-\frac{\sqrt{3}}{2}}{1} = -\frac{\sqrt{3}}{2} \end{align*}

### Concept Problem Solution

Since you know how to measure reference angles now, upon examination of the spinners, you know that the first angle is 30\begin{align*}30^\circ\end{align*}, the second angle is 45\begin{align*}45^\circ\end{align*}, and the third angle is 60\begin{align*}60^\circ\end{align*}.

### Practice

1. Graph 100\begin{align*}100^\circ\end{align*} and identify its reference angle.
2. Graph 200\begin{align*}200^\circ\end{align*} and identify its reference angle.
3. Graph 290\begin{align*}290^\circ\end{align*} and identify its reference angle.

Calculate each value using the unit circle and special right triangles.

1. sin120\begin{align*}\sin 120^\circ\end{align*}
2. cos120\begin{align*}\cos 120^\circ\end{align*}
3. csc120\begin{align*}\csc 120^\circ\end{align*}
4. cos135\begin{align*}\cos 135^\circ\end{align*}
5. sin135\begin{align*}\sin 135^\circ\end{align*}
6. tan135\begin{align*}\tan 135^\circ\end{align*}
7. sin210\begin{align*}\sin 210^\circ\end{align*}
8. cos210\begin{align*}\cos 210^\circ\end{align*}
9. cot210\begin{align*}\cot 210^\circ\end{align*}
10. sin225\begin{align*}\sin 225^\circ\end{align*}
11. cos225\begin{align*}\cos 225^\circ\end{align*}
12. sec225\begin{align*}\sec 225^\circ\end{align*}

### Vocabulary Language: English

Reference Angle

Reference Angle

A reference angle is the angle formed between the terminal side of the angle and the closest of either the positive or negative x-axis.

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Date Created:
Sep 26, 2012