Chapter 10 Trigonometric functions - Ugrad.math.ubc.ca

Math 102 Notes Chapter 10
θ
s
Figure 10.1: The angle θ in radians is related in a simple way to the radius R of the circle, and the
length of the arc S shown.
From Figure 10.1 we see that there is a correspondence between the angle (θ) subtended in a
circle of given radius and the length of arc along the edge of the circle. For a circle of radius R and
angle θ we will define the arclength, S by the relation
S = Rθ
where θ is measured in a convenient unit that we will now select. We now consider a circle of radius
R = 1 (called a unit circle) and denote by s a length of arc around the perimeter of this unit circle.
In this case, the arc length is
S = Rθ = θ
We note that when S = 2π, the arc consists of the entire perimeter of the circle. This leads us to
define the unit called a radian: we will identify an angle of 2π radians with one complete revolution
around the circle. In other words, we use the length of the arc in the unit circle to assign a numerical
value to the angle that it subtends.
We can now use this choice of unit for angles to assign values to any fraction of a revolution,
and thus, to any angle. For example, an angle of 90 ◦ corresponds to one quarter of a revolution
around the perimeter of a unit circle, so we identify the angle π/2 radians with it. One degree is
1/360 of a revolution, corresponding to 2π/360 radians, and so on.
To summarize our choice of units we have the following two points:
1. The length of an arc along the perimeter of a circle of radius R subtended by an
angle θ is S = Rθ where θ is measured in radians.
2. One complete revolution, or one full cycle corresponds to an angle of 2π radians
It is easy to convert between degrees and radians if we remember that 360 ◦ corresponds to 2π
radians. For example, 180 ◦ then corresponds to π radians, 90 ◦ to π/2 radians, etc.
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