What we learn in this section:

Radian Measure 

 
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
2π radians=180
and
1 radian=180π degrees 57.296
or

1=π180 radians0.017 radians.

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 π/4 radians because
π18045=π4.

 

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

sin10.841471butsin10.017452

and when we talk about the angle π/6, we mean π/6 radians (which is 30○ ) not  π/6 ( 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 θ radians on a circle of
radius r, then
s=rθ(θ in radians).
Figure 2: An arc intercepted by the central angle θ.

 

The length s of the arc is the same fraction of the circumference 2πr of the circle that the angle θ is of one complete rotation 2π radians:
s2πr=θ2π. Solving this equation for s, we obtain
s=rθ.

 

Example 1
Find the length of the arc intercepted by a central angle of  50 on a circle of radius 36 in.
Solution
First, we need to express the central angle in radians:
θ=π18050=5π18 rad
Using  Theorem 1, the length of the arc is:
s=rθ=365π18=10π31.4in

 

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 θ is given by:A=12r2θwhere θ 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 πr2 by the ration of the angle and 2π (because the area of the sector is proportional to its angle, and 2π is the angle for the whole circle, in radians):

A=πr2θ2π=12r2θ.

 

 

Example 2

For a circle of radius 18 in, find the area of a sector intercepted by a central angle of 70

Solution

Like the previous example, first, we need to convert the unit of the central angle from degrees to radians:

θ=π18070=7π18

The area is thus

A=12r2θ=12182(7π18)=63π197.9 in2

 

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 θ>2π or θ<2π.

 

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, π/4,315and 405 are first quadrant angles (Figure 5(b)), 5π/6 is a second quadrant angle (Figure 5(c)), and 60 and 300 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

Figure 5