Radians and Degrees

FIELDS OF STUDY: Classical Mechanics; Electronics; Harmonics

ABSTRACT: From the development of astronomy by ancient civilizations to the internal combustion engine, rotations about a center have been an integral part of the development of scientific theories and advances in engineering. Though its origins are not known exactly, the degree as a unit of measurement was originally developed by Babylonians and Persians. The concept of the radian was developed in the early eighteenth century by British mathematician Roger Cotes. This article explains these basic units of angular separation.

Principal Terms

• angle: the separation along a curved path between two rays originating at the same point.
• arclength: the length of an arc defined by an angle and two radii.
• circumference: the distance around a circle defined by a radius.
• cosine: in a right triangle, the ratio of the length of the side adjacent to an acute angle to the length of the hypotenuse.
• pi: the ratio of the circumference of a circle to its diameter.
• radius: distance from the center of a circle to any point along the circle.
• sine: in a right triangle, the ratio of the length of the side opposite an acute angle to the length of the hypotenuse.
• tangent: in a right triangle, the ratio of the length of the side opposite an acute angle to the length of the side adjacent to the same angle.

Origins of Degrees and Radians

For centuries, scientists and mathematicians used the unit of degrees (°) to measure the size of an angle. The origin of the degree is not exactly known. One theory is that it originated in ancient astronomy. Previous civilizations started to notice that the positions of the stars were the same every 360 days—very close to a year. To represent a year, Babylonian and Persian mathematicians divided the circumference of a circle into 360 spaces or units. This unit became known in ancient Greece as the moira, which was translated into Arabic as daraja, meaning a step on a scale or ladder. In Latin, this became de gradus, and in English, degree. With the development of trigonometry, the mathematics of angles, mathematicians and scientists found they needed a way to represent angular measurements as numeric values without arbitrarily determined units such as degrees.

In 1714, British mathematician Roger Cotes (1682–1716) published a paper titled "Logometria." In it, he developed the idea of an angular unit derived from the ratio of an arclength along the circumference of a circle to the radius of the circle. He defined this unit, now known as the radian, as the angle corresponding to an arclength that is equal to the radius of the circle. This measurement of angular separation is approximately equal to 57.3 degrees. The unit of radians uses no arbitrary definitions as to what it means. It can therefore easily be used in mathematical applications ranging from trigonometry to calculus. In trigonometry, the basic functions of the sine, cosine, and tangent of an angle are defined as the ratios between the lengths of two sides of a right triangle. Since the radian is a ratio of two lengths, it has no units and can be used in trigonometry. In calculus, the need for a unit-less quantity was clear when trying to calculate simple harmonic motions of molecules and other simple oscillators.

Calculating Radians

The circumference of a circle is equal to 2πr, where r is the radius of the circle and π is pi, a mathematical constant approximately equal to 3.14. Because a radian corresponds to the central angle of an arclength equal to one radius, there are 2π radians in a circle. A full circle is 360 degrees around. For a circle with a radius equal to one, therefore, one radian is equal to 360/2π degrees, or 180/π degrees. Conversely, one degree is equal to π/180 radians. Multiply the measurement of an angle in degrees (θ_d) by π/180 to convert the angle to radians (θ_r):

For example, if an angle measures 120°, that same angle in radians (rad) is

To convert radians to degrees, simply multiply by 180/π:

For example, if an angle measures π/2 rad, that angle in degrees is

Sample Problem

Consider two angles, one 125° and the other 5π/7 rad. What are the values of the angles in radians and degrees, respectively, given that π has an approximate value of 3.14?

Answer:

To calculate the first angle in radians, multiply 125° by π/180:

The first angle measures approximately 2.18 radians.

To calculate the second angle in degrees, multiply 5π/7 rad by 180/π:

The second angle measures approximately 128.57 degrees.

Applications of Radians and Degrees

The radian and degree both remain useful units of measurement. The degree is still used in navigation and astronomy. It is also used by the general public to explain rotations and revolutions. On the other hand, the radian holds a prominent role in scientific and mathematic developments. It is the International System of Units (SI) derived unit of angular measurement. Without the use of radians, scientists would not be able to properly calibrate electric generators and the angular frequencies at which they spin.

Bibliography

Giambattista, Alan, and Betty McCarthy Richardson. Physics. 2nd ed. Dubuque: McGraw, 2010. Print.

Hemphill, Boyd E., and John C. Polking. "Degree/Radian Circle." Math.Rice.edu. Polking, 4 May 1998. Web. 21 July 2015.

"Introduction to Radians." Khan Academy. Khan Acad., 2015. Web. 21 July 2015.

Joyce, David E. "Dave’s Short Trig Course: Measurement of Angles." Clark University. Dept. of Mathematics and Computer Science, Clark U, 2013. Web. 21 July 2015.

"Measuring Angles in Degrees." Khan Academy. Khan Acad., 2015. Web. 21 July 2015.

Weisstein, Eric W. "Degree." Wolfram MathWorld. Wolfram Research, 1999–2015. Web. 21 July 2015.

Young, Hugh D., and Francis Weston Sears. Sears & Zemansky’s College Physics. 9th ed. Boston: Addison, 2012. Print.