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Practice Conversion between Degrees and Radians

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Introduction to the Unit Circle and Radian Measure

An oddly-shaped house in Asia is built at a 135\begin{align*}135^\circ\end{align*} angle. How many radians is this angle equal to?

The unit circle is the circle centered at the origin with radius equal to one unit. This means that the distance from the origin to any point on the circle is equal to one unit.

Using the unit circle, we can define another unit of measure for angles, radians. Radian measure is based upon the circumference of the unit circle. The circumference of the unit circle is 2π\begin{align*}2 \pi\end{align*} (2πr\begin{align*}2 \pi r\end{align*}, where r=1\begin{align*}r=1\end{align*}). So a full revolution, or 360\begin{align*}360^\circ\end{align*}, is equal to 2π\begin{align*}2 \pi\end{align*} radians. Half a rotation, or 180\begin{align*}180^\circ\end{align*} is equal to π\begin{align*}\pi\end{align*} radians.

One radian is equal to the measure of θ\begin{align*}\theta\end{align*}, the rotation required for the arc length intercepted by the angle to be equal to the radius of the circle. In other words the arc length is 1 unit for θ=1\begin{align*}\theta=1\end{align*} radian.

We can use the equality, π=180\begin{align*}\pi=180^\circ\end{align*} to convert from degrees to radians and vice versa.

To convert from degrees to radians, multiply by π180\begin{align*}\frac{\pi}{180^\circ}\end{align*}.

To convert from radians to degrees, multiply by 180π\begin{align*}\frac{180^\circ}{\pi}\end{align*}.

Let's convert the following units of measure.

1. Convert 250\begin{align*}250^\circ\end{align*} to radians.

To convert from degrees to radians, multiply by π180\begin{align*}\frac{\pi}{180^\circ}\end{align*}. So, 250π180=25π18\begin{align*}\frac{250 \pi}{180}=\frac{25 \pi}{18}\end{align*}.

1. Convert 3π\begin{align*}3 \pi\end{align*} to degrees.

To convert from radians to degrees, multiply by 180π\begin{align*}\frac{180^\circ}{\pi}\end{align*}. So, 3π×180π=3×180=540\begin{align*}3 \pi \times \frac{180^\circ}{\pi}=3 \times 180^\circ=540^\circ\end{align*}.

Now, let's find two angles, one positive and one negative, coterminal to 5π3\begin{align*}\frac{5 \pi}{3}\end{align*} and find its reference angle, in radians.

Since we are working in radians now we will add/subtract multiple of 2π\begin{align*}2 \pi\end{align*} instead of 360\begin{align*}360^\circ\end{align*}. Before we can add, we must get a common denominator of 3 as shown below.

5π3+2π=5π3+6π3=11π3and5π32π=5π36π3=π3\begin{align*}\frac{5 \pi}{3} + 2 \pi = \frac{5 \pi}{3} + \frac{6 \pi}{3} = \frac{11 \pi}{3} \quad and \quad \frac{5 \pi}{3} - 2 \pi = \frac{5 \pi}{3} - \frac{6 \pi}{3} =- \frac{\pi}{3}\end{align*}

Now, to find the reference angle, first determine in which quadrant 5π3\begin{align*}\frac{5 \pi}{3}\end{align*} lies. If we think of the measures of the angles on the axes in terms of π\begin{align*}\pi\end{align*} and more specifically, in terms of π3\begin{align*}\frac{\pi}{3}\end{align*}, this task becomes a little easier.

Consider π\begin{align*}\pi\end{align*} is equal to 3π3\begin{align*}\frac{3 \pi}{3}\end{align*} and 2π\begin{align*}2 \pi\end{align*} is equal to 6π3\begin{align*}\frac{6 \pi}{3}\end{align*} as shown in the diagram. Now we can see that the terminal side of 5π3\begin{align*}\frac{5 \pi}{3}\end{align*} lies in the fourth quadrant and thus the reference angle will be:

6π35π3=π3\begin{align*}\frac{6 \pi}{3} - \frac{5 \pi}{3} = \frac{\pi}{3}\end{align*}

Finally, let's find two angles coterminal to 7π6\begin{align*}\frac{7 \pi}{6}\end{align*}, one positive and one negative, and find its reference angle, in radians.

This time we will add multiples of 2π\begin{align*}2 \pi\end{align*} with a common denominator of 6, or 2π1×66=12π6\begin{align*}\frac{2 \pi}{1} \times \frac{6}{6} = \frac{12 \pi}{6}\end{align*}. For the positive angle, we add to get 7π6+12π6=19π6\begin{align*}\frac{7 \pi}{6} + \frac{12 \pi}{6} = \frac{19 \pi}{6}\end{align*}. For the negative angle, we subtract to get 7π612π6=5π6\begin{align*}\frac{7 \pi}{6} - \frac{12 \pi}{6} = \frac{5 \pi}{6}\end{align*}.

In this case π\begin{align*}\pi\end{align*} is equal to 6π6\begin{align*}\frac{6 \pi}{6}\end{align*} and 2π\begin{align*}2 \pi\end{align*} is equal to 12π6\begin{align*}\frac{12 \pi}{6}\end{align*} as shown in the diagram. Now we can see that the terminal side of 7π6\begin{align*}\frac{7 \pi}{6}\end{align*} lies in the third quadrant and thus the reference angle will be:

7π66π6=π6\begin{align*}\frac{7 \pi}{6} - \frac{6 \pi}{6} = \frac{\pi}{6}\end{align*}

Examples

Example 1

Earlier, you were asked to convert 135\begin{align*}135^\circ\end{align*} to radians.

To convert from degrees to radians, multiply by π180\begin{align*}\frac{\pi}{180^\circ}\end{align*}. So, 135π180=3π4\begin{align*}\frac{135 \pi}{180}=\frac{3 \pi}{4}\end{align*}.

Convert the following angle measures from degrees to radians.

Example 2

45\begin{align*}-45^\circ\end{align*}

45×π180=π4\begin{align*}-45^\circ \times \frac{\pi}{180^\circ}=-\frac{\pi}{4}\end{align*}

Example 3

120\begin{align*}120^\circ\end{align*}

120×π180=2π3\begin{align*}120^\circ \times \frac{\pi}{180^\circ}=\frac{2 \pi}{3}\end{align*}

Example 4

\begin{align*}330^\circ\end{align*}

\begin{align*}330^\circ \times \frac{\pi}{180^\circ}=\frac{11 \pi}{6}\end{align*}

Convert the following angle measures from radians to degrees.

Example 5

\begin{align*}\frac{5 \pi}{6}\end{align*}

\begin{align*}\frac{5 \pi}{6} \times \frac{180^\circ}{\pi}=150^\circ\end{align*}

Example 6

\begin{align*}\frac{13 \pi}{4}\end{align*}

\begin{align*}\frac{13 \pi}{4} \times \frac{180^\circ}{\pi}=585^\circ\end{align*}

Example 7

\begin{align*}-\frac{5 \pi}{2}\end{align*}

\begin{align*}-\frac{5 \pi}{2} \times \frac{180^\circ}{\pi}=-450^\circ\end{align*}

Example 8

Find two coterminal angles to \begin{align*}\frac{11 \pi}{4}\end{align*}, one positive and one negative, and its reference angle.

There are many possible coterminal angles, here are some possibilities:

positive coterminal angle: \begin{align*}\frac{11 \pi}{4} + \frac{8 \pi}{4} = \frac{19 \pi}{4}\end{align*} or \begin{align*}\frac{11 \pi}{4} - \frac{8 \pi}{4} = \frac{3 \pi}{4}\end{align*},

negative coterminal angle: \begin{align*}\frac{11 \pi}{4} - \frac{16 \pi}{4} = -\frac{5 \pi}{4}\end{align*} or \begin{align*}\frac{11 \pi}{4} - \frac{24 \pi}{4} = -\frac{13 \pi}{4}\end{align*}

Using the coterminal angle, \begin{align*}\frac{3 \pi}{4}\end{align*}, which is \begin{align*}\frac{\pi}{4}\end{align*} from \begin{align*}\frac{4 \pi}{4}\end{align*}. So the terminal side lies in the second quadrant and the reference angle is \begin{align*}\frac{\pi}{4}\end{align*}.

Review

For problems 1-5, convert the angle from degrees to radians. Leave answers in terms of \begin{align*}\pi\end{align*}.

1. \begin{align*}135^\circ\end{align*}
2. \begin{align*}240^\circ\end{align*}
3. \begin{align*}-330^\circ\end{align*}
4. \begin{align*}450^\circ\end{align*}
5. \begin{align*}-315^\circ\end{align*}

For problems 6-10, convert the angle measure from radians to degrees.

1. \begin{align*}\frac{7 \pi}{3}\end{align*}
2. \begin{align*}-\frac{13 \pi}{6}\end{align*}
3. \begin{align*}\frac{9 \pi}{2}\end{align*}
4. \begin{align*}-\frac{3 \pi}{4}\end{align*}
5. \begin{align*}\frac{5 \pi}{6}\end{align*}

For problems 11-15, find two coterminal angles (one positive, one negative) and the reference angle for each angle in radians.

1. \begin{align*}\frac{8 \pi}{3}\end{align*}
2. \begin{align*}\frac{11 \pi}{4}\end{align*}
3. \begin{align*}-\frac{\pi}{6}\end{align*}
4. \begin{align*}\frac{4 \pi}{3}\end{align*}
5. \begin{align*}-\frac{17 \pi}{6}\end{align*}

To see the Review answers, open this PDF file and look for section 13.6.

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