Radians, Arcs, and Everything Between

- Trigonometry Angles

In everyday life when most people think of angles and their measurements, they think of a measurement in degrees. However, as previously mentioned, this unit of measurement is based on a crude likening of rotations around a point to the rotation of the earth around the sun. This results in some rather complicated formulas when working with degrees. For this reason, another type of measurement was divised, one that is based in mathmatics. This type of measurement will form the basis of this article in the series on trigonometry.

Radians

Every type of measurement has a base unit because thats how things are measured, by determining the number of base units are in them. In the last post, the unit by which angles were measured was a degree, the rough equivalent of a day. This is an arbitrary unit type though, so what would make a good base unit for measuring angles? To answer that, a review of what an angle is can be illuminating. It's the rotation of one ray away from another. If the outer point of the rotating ray of an angle in standard position is traced for one complete rotation, it forms a circle. Circles formed this way that have a radius of 1 are known as unit circles as shown in Figure 1.

Unit circle

To measure an angle in radians then, is simply to measure the directed length of the intercept arc on this unit circle. The directed length is determined positive or negative by the direction of rotation of the terminal side, positive if counter-clockwise and false if clockwise.

Directed length

In reality, circles of any size radius can be used to determine the radian measure, it just happens that because a unit circle has a length of 1 the resultant calculations are easier. The key is understanding the relationship of the radius of the given circle and the intercept arc of an angle centered in it. By dividing the length of the intercept arc by the radius of the circle, the radian measurement of an angle can be found on any circle.

So angles have a relationship with the radius of a circle, what relationship exists then with the circumference of a circle? As it happens, the relationship has already been identified, the intercept arc of an angle centered on a circle is a portion of the circle's circumference! The circumference of a circle with radius \( r \) is \( 2\pi r \); on a unit circle that has a radius of 1 this is just \( 2\pi \). Therefore, a 360° angle, or an angle whose intercept arc is the complete circle, has a radian measurement of \( 2\pi \), \( 360°\ =\ 2\pi \).

Converting Between Degrees and Radians

From here its a hop, skip and a jump away to the conversion ratio between degrees and radians, simply divide both sides by 2.

\[ 180°\ =\ \pi \]

This gives two different conversion ratios:

\[ \frac{180\ \text{deg}}{\pi\ \text{rad}}\text{ or }\frac{\pi\ \text{rad}}{180\ \text{deg}} \]

Now, it's a simple matter to convert between radians and degrees: just select the conversion ratio that allows the proper units to cancel out.

\[ \require{cancel} 1\ \text{deg}\ =\ \frac{1\ \cancel{\text{deg}}}{1}\ *\ \frac{\pi\ \text{rad}}{180\ \cancel{\text{deg}}}\ =\ \frac{\pi}{180}\text{rad}\ \approx 0.0175\ \text{rad} \]

\[ 1\ \text{rad}\ =\ \frac{1\ \cancel{\text{rad}}}{1}\ *\ \frac{180\ \text{deg}}{\pi\ \cancel{\text{rad}}}\ =\ \frac{180}{\pi}\text{deg}\ \approx 57.3\ \text{deg} \]

This figure shows common angles and their radian/degree measures.

Degrees and radians

Arc Lengths

Some interesting relationships have been identified this far between central angles and the circles they are inscribed within. For angles that exactly 1 complete rotation of the circle, the arc length is the same as the circumference. It has also been established that in radians this can be expressed as \( 2\pi \) and in degrees as 360°. These three relationships can be expressed as follows:

\[ \frac{\text{arc length}}{\text{circumference}} = \frac{m(\alpha)\text{ in rads}}{2\pi\ \text{rad}} = \frac{m(\alpha)\text{ in deg}}{360°\ \text{deg}} \]

For now, the interest is only in the relationship of radian measurements, though the same principles will apply equally to degree measurements. How can this relationship be used to determine the arc length of a central angle? To start, fill in the information that is already known. With the circumference of a circle equaling \( 2\pi r \) and letting \( s \) equal the arc length:

\[ \frac{s}{2\pi r} = \frac{\alpha}{2\pi} \]

Finding the arc length then, is simply solving the above for \( s \). To do this, simply multiply both sides by \( 2\pi r \):

\[ s = \alpha r \]

The arc formula can be used to find the radian measure of an angle from the arc length and circle radius, as well as the circle radius from the arc length and radian measure.

Area of a Sector

The portion of a circle cut out by a central angle is known as a sector. Much like the arc length, the area of a sector is directly proportional to the measure of the central angle. This relationship is shown below:

\[ \frac{\text{area of sector}}{\text{area of circle}} = \frac{m(\alpha)\text{ in rads}}{2\pi\ \text{rad}} = \frac{m(\alpha)\text{ in deg}}{360°\ \text{deg}} \]

Again, the interest here is only in the relationship with radians. With this information, the same strategy that was used to find the arc length can be employed. Given that the area of a circle is \( \pi r^2 \), let \( A \) equal the area of the sector:

\[ \frac{A}{\pi r^2} = \frac{\alpha}{2\pi} \]

Now, solving for A will produce a formula for finding the area of a sector of a circle. This is as simple as multiplying both sides by \( \pi r^2 \):

\[ A = \frac{\alpha r^2}{2} \]

This equation can be used to find the area of a sector from the radian measure of an angle, and vice-versa.

Conclusion

That wraps up the topics of radians, arc lengths, and the areas of sectors. Next up in the series will be an investigative look at angular and linear velocities.