So far, we have worked with right triangles where the largest angle is \(90\ ^{\circ}\). We are now going to extend this to work with angles of any size. This is particularly helpful when working with circular motion, graphing, solving equations, etc.
An angle drawn in standard position on the coordinate plane has its vertex at the origin and the initial side on the positive x-axis as shown below. The terminal side tells us where the angle lies. If it is a positive angle, we draw it by moving in a counterclockwise direction. If it is a negative angle, we draw it by moving in a clockwise direction. We are not restricted to angles between \(0\ ^{\circ}\) and \(360\ ^{\circ}\) since we can keep rotating as many times as necessary.
So far, we have worked with right triangles where the largest angle is \(90\ ^{\circ}\). We are now going to extend this to work with angles of any size. This is particularly helpful when working with circular motion, graphing, solving equations, etc.
An angle drawn in standard position on the coordinate plane has its vertex at the origin and the initial side on the positive x-axis as shown below. The terminal side tells us where the angle lies. If it is a positive angle, we draw it by moving in a counterclockwise direction. If it is a negative angle, we draw it by moving in a clockwise direction. We are not restricted to angles between \(0\ ^{\circ}\) and \(360\ ^{\circ}\) since we can keep rotating as many times as necessary.
Example 1: Draw the following angles in standard position.
a) \(205^{\circ}\) b) \(-342^{\circ}\) c) \(510^{\circ}\)
In order to simplify our work with angles in different quadrants or with angles that exceed one rotation, we will often work with the reference angle. A reference angle is the acute angle that is formed from the terminal side of an angle and the \(x\)-axis. In order to find a reference angle, we often work with the distance that the angle is from \(180\ ^{\circ}\) and \(360\ ^{\circ}\). Below are some examples of finding a reference angle.
Example 2: Draw the following angles in standard position and determine the reference angle.
a) \(298^{\circ}\) b) \(-120^{\circ}\) c) \(27^{\circ}\)
When we were drawing angles above, if an angle was more than \(360\ ^{\circ}\) or less than \(-360\ ^{\circ}\) we kept making as many full rotations as needed before we got to an angle that was in the interval \(0^{\circ}\le\theta\le360^{\circ}\) (or that same interval with negative angles). Essentially, we were finding co-terminal angles. Co-terminal angles are angles that have the same terminal side, and the difference between these angles is always some integer multiple of \(360\ ^{\circ}\).
For example, if we wanted to find some co-terminal angles to \(510\ ^{\circ}\). We can either add or subtract \(360\ ^{\circ}\) to \(510\ ^{\circ}\) as many times as we want to get co-terminal angles. \(510^{\circ}-360^{\circ}=150^{\circ}\) \(150^{\circ}-360^{\circ}=-210^{\circ}\) Both \(150\ ^{\circ}\) and \(-210\ ^{\circ}\) would be co-terminal to \(510\ ^{\circ}\). There are endless other possibilities as well. |
We know that \(180\ ^{\circ}\) is equivalent to \(\pi\) radians. Using this fact, there are several ways that you can convert between degrees and angles, or vice versa.
Converting between degrees and radians
Method One
Degrees to Radians: multiply the angle in degrees by \(\Large{\frac{\pi}{180^{\circ}}}\)
Radians to Degrees: multiply the angle in radians by \(\Large{\frac{180^{\circ}}{\pi}}\)
Method Two
Using Proportions: \(\Large{\frac{D}{180^{\circ}}=\frac{R}{\pi}}\)
Method One
Degrees to Radians: multiply the angle in degrees by \(\Large{\frac{\pi}{180^{\circ}}}\)
Radians to Degrees: multiply the angle in radians by \(\Large{\frac{180^{\circ}}{\pi}}\)
Method Two
Using Proportions: \(\Large{\frac{D}{180^{\circ}}=\frac{R}{\pi}}\)
Example 3: Convert each of the following angles from degrees to radians or radians to degrees. Let’s do the first two using Method One and the second two using Method Two.
a) \(280\ ^{\circ}\)
\(280^{\circ }\cdot \large\frac{\pi }{180^{\circ }}=\frac{280\pi }{180}=\frac{14\pi }{9}\)
b) \(\large-\frac{29\pi }{10}\)
\(\large-\frac{29\pi}{10}\cdot\frac{180^{\circ}}{\pi}=-\frac{29\left(180^{\circ}\right)}{10}=\normalsize-29\left(18^{\circ}\right)=-522^{\circ}\)
c) \(-152\ ^{\circ}\)
\(\large-\frac{152^{\circ}}{180^{\circ}}=\frac{R}{\pi}\)
\(180^{\circ}\left(R\right)=-152^{\circ}\pi\)
\(R=\large-\frac{152^{\circ}\pi}{180^{\circ}}=-\frac{38\pi}{45}\)
d) \(\large\frac{7\pi }{5}\)
\(\large\frac{D}{180^{\circ}}=\Large\frac{\frac{7\pi}{5}}{\pi}\)
\(\pi\left(D\right)=180^{\circ}\left(\frac{7\pi}{5}\right)\)
\(D=180^{\circ}\left(\frac{7}{5}\right)=252^{\circ}\)
Common Degree/Radian Conversions
There are many common degree/radian conversions that we will see regularly. The video below describes what these common conversions are.
This chart shows us common degree/radian conversions. You may use this as a reference, but it's important to be able to find these conversions on your own using the two methods described above.