Radians are a unit of angular measure used in mathematics and physics. One radian is the angle subtended at the center of a circle by an arc whose length is equal to the circle's radius.
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There are $2\pi$ radians in a full circle or $360^\circ$.
Radians provide a direct relationship between the arc length and radius, making calculations in circular motion more straightforward.
In equations involving angular velocity, radians per second ($rad/s$) is the standard unit.
Radians are dimensionless units, often simplifying mathematical expressions in physics.
The conversion from degrees to radians is given by multiplying degrees by $\frac{\pi}{180}$.
Review Questions
How many radians are there in a full circle?
What is the relationship between arc length, radius, and radians?
Convert $90^\circ$ to radians.
Related terms
Angular Velocity: The rate of change of the rotation angle per unit time, typically measured in radians per second ($rad/s$).
Arc Length: The distance along the curved path of a circle; it can be calculated using $s = r \theta$, where $s$ is the arc length, $r$ is the radius, and $\theta$ is the angle in radians.
Rotation Angle: The angle through which an object rotates around a fixed axis; it can be measured in degrees or radians.