In trigonometry, we use the unit “degree” (notation: ^\circ) to measure angles. However, in trigonometric functions, we use radians (notation: rad). Measuring angles using degrees is called the degree measure, while using radians is called the radian measure. This article explains the key points to help you smoothly transition from degree measure to radian measure.
1 Units of Angle (Degrees and Radians)
1.1 Degrees (°) (degree)
You are likely familiar with measuring angles as 45^\circ or 60^\circ. A full circle measures 360^\circ. This method of expressing angles using the symbol ^\circ is known as the degree measure.
1.2 Radians (rad) (radian)
On the other hand, the radian measure expresses angles using radians. Here, the angle is represented as the ratio of the length of an arc to the radius of the circle. More specifically, consider a circle with radius r. From point A(r, 0), take point B so that the arc length between A and B is l. The angle \theta is then defined as the ratio \frac{l}{r} expressed in radians (Figure 1). In other words, the angle \theta in radians is set as shown in Definition 1. This approach to defining angles is what we call the radian measure.
Definition 1 (Definition of an Angle Using Radians) \theta = \frac{l}{r}.
Here, l represents the length of the arc, and r is the radius of the circle.
For example, from the definition of the constant pi (Note 1), we know that c = 2 \pi r (c: circumference) holds true. Considering the arc length l = c, the following calculation applies:
\begin{aligned} \theta &= \frac{l}{r}\\ &= \frac{c}{r}\\ &= \frac{2 \pi r}{r}\\ &= 2 \pi. \end{aligned}
This shows us that the angle corresponding to a full circle’s circumference is 2 \pi radians. In degrees, the full circle corresponds to 360^\circ, so 360^\circ and 2 \pi radians are equivalent angles (Tip 1).
Note 1: Definition of Pi
The constant \pi is defined by the equation:
\pi = \frac{c}{d}.
Here, c is the circumference and d is the diameter of the circle. This value remains constant regardless of the circle’s radius.
2 Conversion Between Degrees and Radians
Using this relationship, let us convert 90^\circ into radians. Since 90^\circ represents \frac{90}{360} = \frac{1}{4} of 360^\circ, multiplying this fraction by 2 \pi gives the equivalent angle in radians:
2 \pi \times \frac{1}{4} = \frac{\pi}{2}.
Thus, 90^\circ in degrees equals \frac{\pi}{2} radians.
Next, consider 60^\circ. As before, find its fraction of 360^\circ, which is \frac{60}{360}. Multiplying by 2 \pi yields:
2 \pi \times \frac{60}{360} = \frac{\pi}{3}.
Therefore, 60^\circ corresponds to \frac{\pi}{3} radians.
Understanding that 360^\circ corresponds to 2 \pi radians allows you to convert easily between degrees and radians. Keep in mind, you do not need to memorise the angle conversions for each specific degree value.
Tip 1: Relationship Between Degrees and Radians
360^\circ \Leftrightarrow 2 \pi
3 Confirmation That Radians Are Dimensionless
Angles measured in radians are defined as a ratio of lengths, as shown in Definition 1 and Note 2. Specifically, radians are obtained by dividing one length by another. Therefore, radians do not have physical units—they are dimensionless quantities. Despite this, for convenience, we call the unit “radian” (rad) to represent angles.
Note 2: Unit Calculations
Units can be multiplied or divided. For example, the equation “distance = speed \times time” involves units:
\begin{aligned} [m] &= [m/s] [s]\\ [m] &= \frac{[m]}{[s]} \times [s]\\ [m] &= [m]. \end{aligned}
The units on the right-hand side simplify to [m], matching the unit on the left-hand side. This shows how units can be consistently manipulated.
4 Summary
We have explained radian measure for angles, contrasting it with degrees. Practice converting smoothly between degrees and radians to acquire confidence. Understanding this correspondence means you do not need to memorise angle sizes in both systems individually.
