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
and
or
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 radians because
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 , we mean radians (which is 30○ ) not ( 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 is the length of an arc of radians on a circle of
radius , then
Figure 2: An arc intercepted by the central angle .
The length of the arc is the same fraction of the circumference of the circle that the angle is of one complete rotation radians:
Solving this equation for , we obtain
Example 1
Find the length of the arc intercepted by a central angle of on a circle of radius in.
Solution
First, we need to express the central angle in radians:
Using Theorem 1, the length of the arc is:
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 of a sector of a circle of radius and central angle is given by:where is measured in radians.
The unit of is the square of the length unit used to measure . For example, if is measured in meters, then will be measured in the square meters.
The area of the sector can be obtained by multiplying the circle’s area by the ration of the angle and (because the area of the sector is proportional to its angle, and is the angle for the whole circle, in radians):
Example 2
For a circle of radius 18 in, find the area of a sector intercepted by a central angle of .
Solution
Like the previous example, first, we need to convert the unit of the central angle from degrees to radians:
The area is thus
Directed Angles
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 or .
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 -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, and are first quadrant angles (Figure 5(b)), is a second quadrant angle (Figure 5(c)), and and 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