1.18: Reference Angles and Angles in the Unit Circle
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.
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James Sousa: Determining Trig Function Values Using Reference Angles and Reference Triangles
Guidance
Consider the angle
Notice that
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.
b.
c.
Solution:
a.
b.
c.
Example B
Find the ordered pair for
Solution:
As we found in Example A, the reference angle for
The terminal side of the angle
Just as the figure above shows
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
Solution:
Using the graph above, you will find that the ordered pair is
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
2. Graph
3. Find the ordered pair for
Solutions:
1. The graph of \begin{align*}210^\circ\end{align*}
and since the angle makes a \begin{align*}30^\circ\end{align*}
2. The graph of \begin{align*}315^\circ\end{align*}
and since the angle makes a \begin{align*}45^\circ\end{align*}
3. Since the reference angle is \begin{align*}30^\circ\end{align*}
\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 \begin{align*}30^\circ\end{align*}
Practice
 Graph \begin{align*}100^\circ\end{align*}
100∘ and identify its reference angle.  Graph \begin{align*}200^\circ\end{align*}
200∘ and identify its reference angle.  Graph \begin{align*}290^\circ\end{align*}
290∘ and identify its reference angle.
Calculate each value using the unit circle and special right triangles.

\begin{align*}\sin 120^\circ\end{align*}
sin120∘ 
\begin{align*}\cos 120^\circ\end{align*}
cos120∘ 
\begin{align*}\csc 120^\circ\end{align*}
csc120∘ 
\begin{align*}\cos 135^\circ\end{align*}
cos135∘ 
\begin{align*}\sin 135^\circ\end{align*}
sin135∘ 
\begin{align*}\tan 135^\circ\end{align*}
tan135∘ 
\begin{align*}\sin 210^\circ\end{align*}
sin210∘ 
\begin{align*}\cos 210^\circ\end{align*}
cos210∘ 
\begin{align*}\cot 210^\circ\end{align*}
cot210∘ 
\begin{align*}\sin 225^\circ\end{align*}
sin225∘ 
\begin{align*}\cos 225^\circ\end{align*}
cos225∘ 
\begin{align*}\sec 225^\circ\end{align*}
sec225∘
Image Attributions
Here you'll learn the definition of reference angles and how to express angles on the unit circle.