radians

(noun)

The angle subtended at the centre of a circle by an arc of the circle of the same length as the circle's radius.

Examples of radians in the following topics:

  • Radians

    • Radians are another way of measuring angles, and the measure of an angle can be converted between degrees and radians.
    • $\displaystyle{ \begin{aligned} 2\pi \text{ radians} &= 360^{\circ} \\ 1\text{ radian} &= \frac{360^{\circ}}{2\pi} \\ 1\text{ radian} &= \frac{180^{\circ}}{\pi} \end{aligned}}$
    • The angle $t$ sweeps out a measure of one radian.
    • (b) An angle of 2 radians has an arc length $s=2r$.
    • Explain the definition of radians in terms of arc length of a unit circle and use this to convert between degrees and radians
  • Angular Position, Theta

    • This result is the basis for defining the units used to measure rotation angles to be radians (rad), defined so that:
    • Because there are 360º in a circle or one revolution, the relationship between radians and degrees is thus 2π rad=360º, so that:
    • Assess the relationship between radians the the revolution of a CD
  • Angular vs. Linear Quantities

    • The units of angular velocity are radians per second.
    • Radian describes the plane angle subtended by a circular arc as the length of the arc divided by the radius of the arc.
    • One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle.
    • More generally, the magnitude in radians of such a subtended angle is equal to the ratio of the arc length to the radius of the circle; that is, $\theta = \frac{s}{r}$, where $\theta$ is the subtended angle in radians, $s$ is arc length, and $r$ is radius.
  • Rotational Angle and Angular Velocity

    • The amount the object rotates is called the rotational angle and may be measured in either degrees or radians.
    • Since the rotational angle is related to the distance $\Delta S$ and to the radius $r$ by the equation $\Delta \theta = \frac{\Delta S}{R}$, it is usually more convenient to use radians.
    • This will give the angular velocity, usually denoted by $\omega$, in terms of radians per second.
  • Period and Frequency

    • Angular frequency is often represented in units of radians per second (recall there are 2π radians in a circle).
    • The locomotive's wheels spin at a frequency of f cycles per second, which can also be described as ω radians per second.
  • Simple Harmonic Motion and Uniform Circular Motion

    • For a path around a circle of radius r, when an angle θ (measured in radians) is swept out, the distance traveled on the edge of the circle is s = rθ.
    • You can prove this yourself by remembering that the circumference of a circle is 2*pi*r, so if the object traveled around the whole circle (one circumference) it will have gone through an angle of 2pi radians and traveled a distance of 2pi*r.
    • The angular velocity ω is in radians per unit time; in this case 2π radians is the time for one revolution T.
  • Introduction to the Polar Coordinate System

    • Angles in polar notation are generally expressed in either degrees or radians ($2\pi $ rad being equal to $360^{\circ}$).
    • Degrees are traditionally used in navigation, surveying, and many applied disciplines, while radians are more common in mathematics and mathematical physics.  
    • The angle $θ$, measured in radians, indicates the direction of $r$. 
    • Adding any number of full turns ($360^{\circ} $ or $2\pi$ radians) to the angular coordinate does not change the corresponding direction.
  • Problem-Solving Techniques

    • Be sure to use units of radians for angles.
    • Before using this equation, we must convert the number of revolutions into radians, because we are dealing with a relationship between linear and rotational quantities:
  • Complex Numbers in Polar Coordinates

    • .$ The other parameter is the angle $\phi$ which the line from the origin to the point makes with the horizontal, measured in radians.
    • So $z$ is the complex number which is $\sqrt2$ units from the origin and whose angle with the horizontal is $\pi/4$ radians which is $45 $ degrees, while $w$ is the number whose distance from the origin is $\sqrt2$ and whose angle with the origin is $3\pi/4$ radians which is $135$ degrees. 
  • Defining Trigonometric Functions on the Unit Circle

    • The angle $t$ (in radians) forms an arc of length $s$.
    • The coordinates of certain points on the unit circle and the the measure of each angle in radians and degrees are shown in the unit circle coordinates diagram.
    • Coordinates of a point on a unit circle where the central angle is $t$ radians.
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