Radians are a unit of angular measure used in mathematics. One radian is the angle created when the radius of a circle is wrapped along its circumference.
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The total angle around a circle is $2\pi$ radians.
There are $\pi$ radians in 180 degrees, making it easy to convert between degrees and radians using the fact that $1 \text{ radian} = \frac{180}{\pi}$ degrees.
Radians provide a direct relationship between the arc length and the radius of a circle, where the arc length is equal to the product of the radius and the angle in radians.
$\sin(\theta)$, $\cos(\theta)$, and other trigonometric functions often use angles measured in radians for calculus operations like differentiation and integration.
$\pi/6$, $\pi/4$, $\pi/3$, $\pi/2$, and $\pi$ are common angles in radians that correspond to familiar degree measures (30°, 45°, 60°, 90°, and 180° respectively).
Review Questions
How many radians are there in a full circle?
Convert 45 degrees to radians.
Why are radians preferred over degrees in calculus for trigonometric functions?
Related terms
Degree: A unit of angular measure where one degree represents $1/360$th of a full rotation.
Unit Circle: A circle with a radius of one centered at the origin of a coordinate plane; commonly used to define trigonometric functions.
$\pi$ (Pi): $\pi$ is an irrational number approximately equal to 3.14159, representing the ratio of any circle's circumference to its diameter.