You may learn about radians for the first time in a trigonometry class. Simply, a radian is another way to measure an angle instead of using degrees. You are merely switching the unit of measurement. It is like measuring your height in centimeters instead of inches.
Definition
In science and engineering, radians are much more convenient (and common) than degrees. A radian is defined as the angle between 2 radii (radiuses) of a circle where the arc between them has length of one radius.
Another way of putting it is: “a radian is the angle subtended by an arc of length r (the radius)”.
One radian is about 57.3∘.
Radians are especially useful in calculus where we want to interchange angles and other quantities (e.g., length). For example, see how radians are required in Fourier Series.
Most computer programs use radians as the default.1
Radians has a canonical definition relative to the geometry of a circle. Degrees cuts a circle into 360 equal angles. Why 360? It has a lot of nice factors so it is usefully divisible. But 360 is a fairly arbitrary number.
A radian, on the other hand, can be defined through the basic definition of the circle. A circle is constructed by choosing a length, the radius, and drawing all the points that are that distance from a central point.
Now mark a point on the circle. Walk along the circumference of the circle a distance equal to the radius. Mark your location. Draw radii from the center to your two points. The angle between those two radii measures 1 radian. Using Greek mathematics, we can prove that a semicircle measures π radians.3
Who
For those who learned about radians in trigonometry class but never had an opportunity to use them or how important they are in mathematics, engineering and physics.
What
The practically of using radian measure is that many of the mathematical expressions involving angle measurement are simpler when using radian measure than when using degrees. Specifically the derivatives of trig functions are much easier to express if the angles are measured in radians rather than degrees. If you are dealing with the rate of change of an angle then you are probably going to need the derivative and hence there is an advantage to using radians.
Dealing with the rate of change of an angle is much more common than you might think. In North America the electricity delivered is to our houses at 60 Hz (60 hertz where one hertz is one cycle per second), middle c is 261.625565 hertz, the tachometer in my car registers in revolutions per minute, the processor in my computer runs at 2.66 GHz. These are all in revolutions or cycles per unit time but the people who work with these quantities more often use the units radians per unit time. One revolution or cycle is 2π radians and when using cycles or revolutoions the factor 2π appears quite often and complicates the expressions and introduces additional round off error in the calculations.
What professionals might use radians? Any engineer or scientist who deals with electricity, someone who works with electronic music, automotive engineers, electronic circuit designers, and my favourite, mathematicians.2
Why
See Theoretical Knowledge Vs Practical Application.
How
Many of the References, Additional Reading, websites and YouTube videos will assist you with your further understanding of radians. As some professors say: “It is intuitively obvious to even the most casual observer.”
References
1 Bourne, Murray. 2021. “7. Radians”. intmath.com. https://www.intmath.com/trigonometric-functions/7-radians.php.
2 “Is There A Practical Use For Radian Measure? – Math Central”. 2021. mathcentral.uregina.ca. http://mathcentral.uregina.ca/QQ/database/QQ.09.07/h/paula2.html.
3 Sklar, Max. “What Is The Purpose Of Radian? Why Do Mathematicians Feel The Need To Divide A Semicircle Into Π Equal Arcs?” 2022. Quora. https://qr.ae/pvFsZo.
Additional Reading
“Intuitive Guide To Angles, Degrees And Radians – BetterExplained”. 2021. betterexplained.com. https://betterexplained.com/articles/intuitive-guide-to-angles-degrees-and-radians/.
McMullin, Lin. 2012. “Why Radians?”. Teaching Calculus. https://teachingcalculus.com/2012/10/12/951/.
Radians make it possible to relate a linear measure and an angle measure. A unit circle is a circle whose radius is one unit. The one unit radius is the same as one unit along the circumference. Wrap a number line counterclockwise around a unit circle starting with zero at (1, 0). The length of the arc subtended by the central angle becomes the radian measure of the angle.
“Radians”. 2021. Digit Math. https://www.digitmath.com/trigonometry-radians.html.
“Radians Vs. Degrees – Physicsthisweek.Com”. 2021. physicsthisweek.com. https://www.physicsthisweek.com/math/radians-vs-degrees/.
You should use radians when you are looking at objects moving in circular paths or parts of circular path. In particular, rotational motion equations are almost always expressed using radians. The initial parameters of a problem might be in degrees, but you should convert these angles to radians before using them.
You should use degrees when you are measuring angles using a protractor, or describing a physical picture. Most people have developed intuitive feel for the common angles. This would be common in vector related problems, including speeds, projectiles, forces, and similar situations.
“What Are “Radians”? A Simple Explanation Of Radian Vs. Degree”. 2021. Medium. https://medium.com/star-gazers/what-are-radians-a-simple-explanation-of-radian-vs-degree-565062445f69.
You may learn about radians for the first time in a trigonometry class. Simply, a radian is another way to measure an angle instead of using degrees. You are merely switching the unit of measurement. It is like measuring your height in centimeters instead of inches.
Videos
In this lesson, we will learn how to use the unit circle to calculate the sin, cos, or tangent of an angle in radians. The first step is to draw the angle and determine which quadrant the angle lies. Next, we find the reference angle, which is the angle from the terminal ray of our angle to the nearest x-axis. Finally, we use the reference angle to find the sine and cosine values in quadrant 1 and insert a negative or positive sign on the front of the answer depending on the quadrant that the angle lies in. In order to calculate the tangent, we simply first find the sine and cosine, the divide the two numbers to calculate the tangent of the angle.
Understanding the definition and motivation for radians and the relationship between radians and degrees.
Mathematical Mysteries 