radian

Examples of radian 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

• 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.

• 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.

• Sine and Cosine as Functions

  • Below are some of the values for the sine function on a unit circle, with the $x$-coordinate being the angle in radians and the $y$-coordinate being $\sin x$:
  • Below are some of the values for the sine function on a unit circle, with the $x$-coordinate being the angle in radians and the $y$-coordinate being $\cos x$:
  • Graph of points with $x$ coordinates being angles in radians, and $y$ coordinates being the function $\sin x$.

• 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.
  • Then $w$ 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. 

• Tangent as a Function

  • Consider the points below, for which the $x$-coordinates are angles in radians, and the $y$-coordinates are $\tan x$:

• Special Angles

  • A reference angle is always an angle between $0$ and $90^{\circ}$, or $0$ and $\displaystyle{\frac{\pi}{2}}$ radians.