# 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.

• 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.