4.1 | Radian and Degree Measure


In a two dimensional Cartesian coordinate system, an angle is typically defined by its two sides, with its vertex at the origin. The initial side is on the positive \(x\)-axis, while the other side or terminal side is defined by the angular measure from the initial side in radians, degrees, or turns. Positive angles represent rotations toward the positive \(y\)-axis and negative angles represent rotations toward the negative \(y\)-axis. When Cartesian coordinates are represented by standard position, defined by the \(x\)-axis rightward and the \(y\)-axis upward, positive rotations are counterclockwise and negative rotations are clockwise.

It is common to use Greek letters \( \alpha, \beta, \gamma, \theta, \phi, …\) to serve as variables standing for the size of some angle. (To avoid confusion with its other meaning, the symbol \(\pi\) is typically not used for this purpose.)

Two angles \(\alpha\) and \(\beta\) are said to be coterminal provided their initial and terminal sides coincide.

We now define a new angle measure called radians. Given a circle of radius \(r\), the amount of rotation in degrees of a point on that circle is directly related to the distance traveled by that point along the circumference of that circle. However, whereas degrees is independent of \(r\) so must be our new angle measure. Dividing the circumference by \(r\) removes this dependency and thereby allows us to establish the relationship
$$
360^\circ = 2 \pi
$$
where the right-hand side is in the new angle measure called radians. Units for this measure are usually suppressed. We can write rad or radians to emphasize the units, if necessary.

As a consequence of the above definition, we see that
\(180^\circ = \pi\), \(90^\circ = \pi/2\), and
$$
1 \text{ radian } = \frac{180}{\pi}^\circ
$$
or
$$
1^\circ = \frac{\pi}{180} \; rad
$$
The last two equation can be used to convert between degrees and radians.

Conversions Between Degrees and Radians

  • To convert degrees to radians, multiply degrees by \(\frac{\pi}{180}\)
  • To convert radians to degrees, multiply degrees by \(\frac{180}{\pi}\)
We’ve defined radians constructively. A more direct and ubiquitous definition follows.
One radian is the measure of a central angle \(\theta\) that intercepts an arc \(s\) equal in length to the radius \(r\) of the circle. That is,
$$
\theta = \frac{s}{r}
$$
where \(\theta\) is measured in radians.
For a circle of radius \(r\), a central angle \(\theta\) intercepts an arc of length \(s\) given by
$$s = r \theta$$
where \(\theta\) is measured in radians.
Consider a particle moving at a constant speed along a circular arc of radius \(r\). If \(s\) is the length of the arc traveled in time \(t\), then linear speed \(v\) of the particle is
$$
v = \frac{s}{t}
$$
Moreover, if \(\theta\) is the angle (in radian measure) corresponding to the arc length \(s\), then the angular speed \(\omega\) of the particle is
$$
\omega = \frac{\theta}{r}
$$
A sector of a circle is the region bounded by two radii of the circle and their intercepted arc.
For a circle of radius \(r\), the area \(A\) of a sector of the circle with central angle \(\theta\) is given by
$$
A = \frac{1}{2} r^2 \theta
$$
where \(\theta\) is measured in radians.

E 4.1 Exercises