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Difficulty Level: At Grade Created by: CK-12

## Introduction

Now that we know how to find the sine, cosine and tangent of any angle, we can extend this concept to the \begin{align*}x-y\end{align*} plane. First, we need to derive a different way to measure angles, called radians. Radians are much like arc length. This way, we can take the “length” of a degree measurement and plot it like \begin{align*}x\end{align*}. Then, the value of the function is the \begin{align*}y\end{align*} value on the graph. In this manner we will be able to see the six trigonometric functions in a whole new way.

## Learning Objectives

• Convert angle measure from degrees to radians and radians to degrees.
• Calculate the values of the 6 trigonometric functions for special angles in terms of radians or degrees.

Until now, we have used degrees to measure angles. But, what exactly is a degree? A degree is \begin{align*}\frac{1}{360^{th}}\end{align*} of a complete rotation around a circle. Radians are alternate units used to measure angles in trigonometry. Just as it sounds, a radian is based on the radius of a circle. One radian (abbreviated rad) is the angle created by bending the radius length around the arc of a circle. Because a radian is based on an actual part of the circle rather than an arbitrary division, it is a much more natural unit of angle measure for upper level mathematics.

What if we were to rotate all the way around the circle? Continuing to add radius lengths, we find that it takes a little more than 6 of them to complete the rotation.

Recall from geometry that the arc length of a complete rotation is the circumference, where the formula is equal to \begin{align*}2\pi\end{align*} times the length of the radius. \begin{align*}2\pi\end{align*} is approximately 6.28, so the circumference is a little more than 6 radius lengths. Or, in terms of radian measure, a complete rotation (360 degrees) is \begin{align*}2\pi\end{align*} radians.

\begin{align*}360 \ \text{degrees} = 2\pi \ \text{radians}\end{align*}

With this as our starting point, we can find the radian measure of other angles. Half of a rotation, or 180 degrees, must therefore be \begin{align*}\pi\end{align*} radians, and 90 degrees must be \begin{align*}\frac{1}{2} \pi\end{align*}, written \begin{align*}\frac{\pi}{2}\end{align*}.

Example 1: Find the radian measure of these angles.

Angle in Degrees Angle in Radians
90 \begin{align*}\frac{\pi}{2}\end{align*}
45
30
60
75

Solution: Because 45 is half of 90, half of \begin{align*}\frac{1}{2} \pi\end{align*} is \begin{align*}\frac{1}{4} \pi\end{align*}. 30 is one-third of a right angle, so multiplying gives:

\begin{align*}\frac{\pi}{2} \times \frac{1}{3}=\frac{\pi}{6}\end{align*}

and because 60 is twice as large as 30:

\begin{align*}2 \times \frac{\pi}{6}=\frac{2 \pi}{6}=\frac{\pi}{3}\end{align*}

Here is the completed table:

Angle in Degrees Angle in Radians
\begin{align*}90\end{align*} \begin{align*}\frac{\pi}{2}\end{align*}
\begin{align*}45\end{align*} \begin{align*}\frac{\pi}{4}\end{align*}
\begin{align*}30\end{align*} \begin{align*}\frac{\pi}{6}\end{align*}
\begin{align*}60\end{align*} \begin{align*}\frac{\pi}{3}\end{align*}

There is a formula to convert between radians and degrees that you may already have discovered while doing this example. However, many angles that are commonly used can be found easily from the values in this table. For example, most students find it easy to remember 30 and 60. 30 is \begin{align*}\pi\end{align*} over 6 and 60 is \begin{align*}\pi\end{align*} over 3. Knowing these angles, you can find any of the special angles that have reference angles of 30 and 60 because they will all have the same denominators. The same is true of multiples of \begin{align*}\frac{\pi}{4}\end{align*} (45 degrees) and \begin{align*}\frac{\pi}{2}\end{align*} (90 degrees).

Extending the radian measure past the first quadrant, the quadrantal angles have been determined, except \begin{align*}270^\circ\end{align*}. Because \begin{align*}270^\circ\end{align*} is halfway between \begin{align*}180^\circ\end{align*}, \begin{align*}\pi\end{align*}, and \begin{align*}360^\circ\end{align*}, \begin{align*}2 \pi\end{align*}, it must be \begin{align*}1.5\pi\end{align*}, usually written \begin{align*}\frac{3\pi}{2}\end{align*}.

For the \begin{align*}45^\circ\end{align*} angles, the radians are all multiples of \begin{align*}\frac{\pi}{4}\end{align*}. For example, \begin{align*}135^\circ\end{align*} is \begin{align*}3 \cdot 45^\circ\end{align*}. Therefore, the radian measure should be \begin{align*}3 \cdot \frac{\pi}{4}\end{align*}, or \begin{align*}\frac{3\pi}{4}\end{align*}. Here are the rest of the multiples of \begin{align*}45^\circ\end{align*}, in radians:

Notice that the additional angles in the drawing all have reference angles of 45 degrees and their radian measures are all multiples of \begin{align*}\frac{\pi}{4}\end{align*}. All of the even multiples are the quadrantal angles and are reduced, just like any other fraction.

Example 2: Complete the following radian measures by counting in multiples of \begin{align*}\frac{\pi}{3}\end{align*} and \begin{align*}\frac{\pi}{6}\end{align*}:

Solution:

Notice that all of the angles with 60-degree reference angles are multiples of \begin{align*}\frac{\pi}{3}\end{align*}, and all of those with 30-degree reference angles are multiples of \begin{align*}\frac{\pi}{6}\end{align*}. Counting in these terms based on this pattern, rather than converting back to degrees, will help you better understand radians.

## Converting Any Degree to Radians

For all examples there is a conversion formula. This formula works for all degrees and radians. Remember that: \begin{align*}\pi \ \text{radians} = 180^\circ\end{align*}. If you divide both sides of this equation by \begin{align*}\pi\end{align*}, you will have the conversion formula:

\begin{align*}\text{radians} \times \frac{180}{\pi}=\text{degrees}\end{align*}

If we have a degree measure and wish to convert it to radians, then manipulating the equation above gives:

\begin{align*}\text{degrees} \times \frac{\pi}{180}=\text{radians}\end{align*}

Example 3: Convert \begin{align*}\frac{11\pi}{3}\end{align*} to degree measure.

From the last section, you should recognize that this angle is a multiple of \begin{align*}\frac{\pi}{3}\end{align*} (or 60 degrees), so there are 11, \begin{align*}\frac{\pi}{3}\end{align*}'s in this angle, \begin{align*}\frac{\pi}{3} \times 11 = 60^\circ \times 11 = 660^\circ\end{align*}.

Here is what it would look like using the formula:

\begin{align*}\text{radians} \times \frac{180}{\pi} = \text{degrees}\end{align*}

Example 4: Convert \begin{align*}-120^\circ\end{align*} to radian measure. Leave the answer in terms of \begin{align*}\pi\end{align*}.

\begin{align*}\text{degrees} \times \frac{\pi}{180}&=\text{radians}\\ -120^\circ \times \frac{\pi}{180}&=\frac{-120^\circ \pi}{180}\end{align*}

and reducing to lowest terms gives us \begin{align*}-\frac{2\pi}{3}\end{align*}

You could also have noticed that 120 is \begin{align*}2 \times 60\end{align*}. Since \begin{align*}60^\circ\end{align*} is \begin{align*}\frac{\pi}{3}\end{align*} radians, then 120 is 2, \begin{align*}\frac{\pi}{3}\end{align*}’s, or \begin{align*}\frac{2\pi}{3}\end{align*}. Make it negative and you have the answer, \begin{align*}-\frac{2\pi}{3}\end{align*}.

Example 5: Express \begin{align*}\frac{11\pi}{12}\end{align*} radians terms of degrees.

\begin{align*}\text{radians} \times \frac{180}{\pi}=\text{degrees}\end{align*}

Note: Sometimes students have trouble remembering if it is \begin{align*}\frac{180}{\pi}\end{align*} or \begin{align*}\frac{\pi}{180}\end{align*}. It might be helpful to remember that radian measure is almost always expressed in terms of \begin{align*}\pi\end{align*}. If you want to convert from radians to degrees, you want the \begin{align*}\pi\end{align*} to cancel out when you multiply, so it must be in the denominator.

## The Six Trig Functions and Radians

Even though you are used to performing the trig functions on degrees, they still will work on radians. The only difference is the way the problem looks. If you see \begin{align*}\sin \frac{\pi}{6}\end{align*}, that is still \begin{align*}\sin 30^\circ\end{align*} and the answer is still \begin{align*}\frac{1}{2}\end{align*}.

Example 6: Find \begin{align*}\tan \frac{3\pi}{4}\end{align*}.

Solution: If needed, convert \begin{align*}\frac{3\pi}{4}\end{align*} to degrees. Doing this, we find that it is \begin{align*}135^\circ\end{align*}. So, this is \begin{align*}\tan 135^\circ\end{align*}, which is -1.

Example 7: Find the value of \begin{align*}\cos \frac{11\pi}{6}\end{align*}.

Solution: If needed, convert \begin{align*}\frac{11\pi}{6}\end{align*} to degrees. Doing this, we find that it is \begin{align*}330^\circ\end{align*}. So, this is \begin{align*}\cos 330^\circ\end{align*}, which is \begin{align*}\frac{\sqrt{3}}{2}\end{align*}.

Example 8: Convert 1 radian to degree measure.

Solution: Many students get so used to using \begin{align*}\pi\end{align*} in radian measure that they incorrectly think that 1 radian means \begin{align*}1\pi\end{align*} radians. While it is more convenient and common to express radian measure in terms of \begin{align*}\pi\end{align*}, don’t lose sight of the fact that \begin{align*}\pi\end{align*} radians is a number. It specifies an angle created by a rotation of approximately 3.14 radius lengths. So 1 radian is a rotation created by an arc that is only a single radius in length.

\begin{align*}\text{radians} \times \frac{180}{\pi}=\text{degrees}\end{align*}

So 1 radian would be \begin{align*}\frac{180}{\pi}\end{align*} degrees. Using any scientific or graphing calculator will give a reasonable approximation for this degree measure, approximately \begin{align*}57.3^\circ\end{align*}.

Example 9: Find the radian measure of an acute angle, \begin{align*}\theta\end{align*}, with \begin{align*}\sin \theta=\frac{\sqrt{2}}{2}\end{align*}.

Solution: Here, we are working backwards. From last chapter, you may recognize that \begin{align*}\frac{\sqrt{2}}{2}\end{align*} goes with \begin{align*}45^\circ\end{align*}. Because the example is asking for an acute angle, we just need to convert \begin{align*}45^\circ\end{align*} to radians. \begin{align*}45^\circ\end{align*} in radians is \begin{align*}\frac{\pi}{4}\end{align*}.

## Check the Mode

Most scientific and graphing calculators have a MODE setting that will allow you to either convert between the two, or to find approximations for trig functions using either measure. It is important that if you are using your calculator to estimate a trig function that you know which mode you are using. Look at the following screen:

If you entered this expecting to find the sine of 30 degrees you would realize based on the last chapter that something is wrong because it should be \begin{align*}\frac{1}{2}\end{align*}. In fact, as you may have suspected, the calculator is interpreting this as 30 radians. In this case, changing the mode to degrees and recalculating will give the expected result.

Scientific calculators will usually have a 3-letter display that shows either DEG or RAD to tell you which mode the calculator is in.

## Points to Consider

• In certain cases, why are radians more useful than degrees?
• Think about the steps you would take to solve \begin{align*}\sin \frac{11\pi}{6}\end{align*}. Are these step similar for finding any trig function for any angle in radians?

## Review Questions

1. The following picture is a sign for a store that sells cheese.
1. Estimate the degree measure of the angle of the circle that is missing.
2. Convert that measure to radians.
3. What is the radian measure of the part of the cheese that remains?
2. Convert the following degree measures to radians. All answers should be in terms of \begin{align*}\pi\end{align*}.
1. \begin{align*}240^\circ\end{align*}
2. \begin{align*}270^\circ\end{align*}
3. \begin{align*}315^\circ\end{align*}
4. \begin{align*}-210^\circ\end{align*}
5. \begin{align*}120^\circ\end{align*}
6. \begin{align*}15^\circ\end{align*}
7. \begin{align*}-450^\circ\end{align*}
8. \begin{align*}72^\circ\end{align*}
9. \begin{align*}720^\circ\end{align*}
10. \begin{align*}330^\circ\end{align*}
3. Convert the following radian measures to degrees:
1. \begin{align*}\frac{\pi}{2}\end{align*}
2. \begin{align*}\frac{11\pi}{5}\end{align*}
3. \begin{align*}\frac{2\pi}{3}\end{align*}
4. \begin{align*}5\pi\end{align*}
5. \begin{align*}\frac{7\pi}{2}\end{align*}
6. \begin{align*}\frac{3\pi}{10}\end{align*}
7. \begin{align*}\frac{5\pi}{12}\end{align*}
8. \begin{align*}-\frac{13\pi}{6}\end{align*}
9. \begin{align*}\frac{8}{\pi}\end{align*}
10. \begin{align*}\frac{4 \pi}{15}\end{align*}
4. The drawing shows all the quadrant angles as well as those with reference angles of \begin{align*}30^\circ\end{align*}, \begin{align*}45^\circ\end{align*}, and \begin{align*}60^\circ\end{align*}. On the inner circle, label all angles with their radian measure in terms of \begin{align*}\pi\end{align*} and on the outer circle, label all the angles with their degree measure.
5. Using a calculator, find the approximate degree measure (to the nearest tenth) of each angle expressed in radians.
1. \begin{align*}\frac{6\pi}{7}\end{align*}
4. \begin{align*}\frac{20\pi}{11}\end{align*}
6. Gina wanted to calculate the \begin{align*}\sin 210^\circ\end{align*} and got the following answer on her calculator: Fortunately, Kylie saw her answer and told her that it was obviously incorrect.
2. Explain what she did wrong.
\begin{align*}x\end{align*} \begin{align*}\text{Sin}(x)\end{align*} \begin{align*}\text{Cos}(x)\end{align*} \begin{align*}\text{Tan}(x)\end{align*}
\begin{align*}\frac{5\pi}{4}\end{align*}
\begin{align*}\frac{11\pi}{6}\end{align*}
\begin{align*}\frac{2\pi}{3}\end{align*}
\begin{align*}\frac{\pi}{2}\end{align*}
\begin{align*}\frac{7\pi}{2}\end{align*}

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