We can obtain a unit for measuring angles by subdividing a right angle into a number of equal parts. Traditionally we subdivide the right angle into 90 equal parts; each part is 1 “degree” (denoted by 1○ ), and thus there are 360 degrees in a complete circle. We could also subdivide the right angle into 100 parts, which brings us the decimal system for angles, but this unit is not common. Alternatively, we can describe the size of an angle by using an essentially different method of measurement, the so-called radian measure. The use of this unit simplifies many formulas in trigonometry and calculus.
To measure an angle in radians, we place its vertex at the center of the unit circle (i.e. circle of radius 1). This angle cuts out an arc on the circumference of this circle. The length of this arc is called the size of the angle in radians (Figure 1). Because the total circumference of the unit circle (note that the radius is 1) is 2 π and one complete rotation is 360○ , we have
Using the above formulas, we can easily convert the angle measure from degrees to radians or from radians to degrees. For example, 45○ is the same as $\pi/4$ radians because
\[\frac{\pi}{180}\cdot45=\frac{\pi}{4}.\]
In mathematics, whenever angles occur, the default is that the angles are measured in radians (instead of degrees) unless you see the degree symbol (○). Therefore if you want to calculate sin 1, you need to set your calculator in radian mode and if you want to find sin 1○, you need to set your calculator in degree mode. Therefore
and when we talk about the angle $\pi/6$, we mean $\pi/6$ radians (which is 30○ ) not $\pi/6$ ( $\approx$ 0.524 ) degrees.
Figure 1
Arc Length
An arc on a circle can be measured by the angle it subtends in radians. What is the length s of this arc, as in Figure 2? The answer is as follows.
Theorem 1 (Arc Length): If $s$ is the length of an arc of $\theta$ radians on a circle of
radius $r$, then
\[
s=r\theta\qquad(\theta\text{ in radians)}.
\]
Figure 2: An arc intercepted by the central angle $\theta$.
The length $s$ of the arc is the same fraction of the circumference $2\pi r$ of the circle that the angle $\theta$ is of one complete rotation $2\pi$ radians:
\[
\frac{s}{2\pi r}=\frac{\theta}{2\pi}.
\]
Solving this equation for $s$, we obtain
\[
s=r\theta.
\]
Example 1
Find the length of the arc intercepted by a central angle of $50^\circ$ on a circle of radius $36$ in.
Solution
First, we need to express the central angle in radians:
\begin{equation*}
\theta=\frac{\pi }{180} 50=\frac{5\pi}{18}\ \text{rad}
\end{equation*}
Using Theorem 1, the length of the arc is:
\begin{align*}
s&=r\theta\\
&=36 \frac{5\pi}{18}\\
&=10\pi\approx 31.4\,\text{in}
\end{align*}
Area of a Sector
A circular sector (or simply a sector) is a portion of a disk enclosed by two radii and an arc (the shaded part of Figure 3):
Figure 3: A circular sector is shaded in purple.
Theorem 2 (Area of a Sector): The area $A$ of a sector of a circle of radius $r$ and central angle $\theta$ is given by:\[A=\frac{1}{2}r^2\theta\]where $\theta$ is measured in radians.
The unit of $A$ is the square of the length unit used to measure $r$. For example, if $r$ is measured in meters, then $A$ will be measured in the square meters.
The area of the sector can be obtained by multiplying the circle’s area $\pi r^2$ by the ratio of the angle and $2\pi$ (because the area of the sector is proportional to its angle, and $2\pi$ is the angle for the whole circle, in radians):
Any angle has two sides. To measure rotations, we designate one side as the initial side and the other as the terminal side. Then we think of angle as being generated by the rotation of the initial side to the final side. If the rotation is counterclockwise, the angle is regarded positive and if the rotation is clockwise, the angle is negative (Figure 4).
We allow angles of more than one complete rotation. That is, we also allow $\theta>2\pi$ or $\theta<-2\pi$.
Angles in standard position
We say the angle is in its standard position if its vertex is at the origin of the coordinate system and its initial side is along the positive $x$-axis.
(a) A positive angle in standard position.
(b) A negative angle in standard position
Figure 4
We say an angle is a first quadrant angle or is in the first quadrant if in standard position, its terminal end lies in the first quadrant. Similar definitions hold for second, third or fourth quadrant angles (Figure 5(a)). For example, $\pi/4, -315^\circ$and $405^{\circ}$ are first quadrant angles (Figure 5(b)), $5\pi/6$ is a second quadrant angle (Figure 5(c)), and $-60^{\circ}$ and $300^{\circ}$ are fourth quadrant angles.
(a) The angle shown here is a second quadrant angle
(b) The angles shown here are first quadrant angles
(c) The angle shown here is a second quadrant angle
(d) The angles shown here are fourth quadrant angles
