Radian Measure

Definition and Basics

Radian is a unit of measurement for angles. Just like degrees, it gives a sense of how “open” an angle is.
One radian is the size of the angle subtended at the centre of a circle by an arc which is equal in length to the radius of the circle.
In simpler terms, if you wrap the radius of a circle along its circumference, the angle you get at the centre is one radian.
The total angle in any circle is 2π radians (equally 360°). Hence, π radians (π rad) is equal to 180°.

Conversion

You can convert between degrees and radian measure by using the relationship that 180° equals π radians.
To convert from radians to degrees, multiply the radians by 180/π.
To convert from degrees to radians, multiply the degrees by π/180.

Applications in Trigonometry

The functions sine, cosine, and tangent (collectively known as the trigonometric functions) take radian measures as inputs in their “pure” form.
For example, the sine of an angle measured in radians gives the y-coordinate of the point on the unit circle that represents the angle.
These functions have special properties when used with radian measure, such as the fact that they are periodic with period 2π in radian measure.

Circular Functions and Radians

The functions that describe motion around a circle (used in physics and engineering) use radian measure.
If you want to describe, say, the angular velocity of an object moving around a circle, you’d use radians in the formula.

Small Angles in Radians

For very small angles, it’s often useful to consider the angle in radians.
This is because the value of a small angle in radians is close to the sine of the angle and to the angle minus the cosine of the angle. This relationship is very useful in a variety of applications.

Arc Length and Radian Measurements

Radian measurements are also very convenient when dealing with arc lengths.
This is because the definition of radian measure directly connects an arc’s length to its subtended angle.
Specifically, the length of an arc subtended by an angle θ (in radians) on a circle of radius r can be calculated as: Arc length = rθ.