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Section 13-2:
Measurement of
Arcs and Rotations
Chapter 13:
Trigonometric and
Circular Functions
Objectives
Given the angle measure, q, or the arc
length, x, sketch the position of the point P
on the unit circle.
Standard Position
A directed angle or arc beginning at the
positive u-axis and measured
counterclockwise is said to be in standard
position.
Thus, if q or x is negative, the angle or arc
would be measured in the clockwise
direction.
Examples
Graph the following angles:
57º
133º
-251º
560º
Coterminal
Angles or arcs in standard position are said
to be coterminal if they terminate (end) at
the same place.
Coterminal arcs or angles differ by an
integral number of revolutions.
Example: 60º and 420º are a pair of
coterminal angles.
Reference Angle
In order to draw an angle or arc in standard
position, it helps to find the measure of the
acute angle between the terminal side and
the u-axis.
This angle is called the reference angle
Finding the Reference Angle
We have formulas for finding the reference
angle, but it is often easier to figure out the
reference angle by drawing a picture.
Reference Angle Formulas:
If q terminates in Quadrant I: qref = q
If q terminates in Quadrant II: qref = 180º – q
If q terminates in Quadrant I: qref = q - 180º
If q terminates in Quadrant IV: qref = 360º – q
Example:
Find the reference angles for the following
angles:
156º
214º
336º
4897º
Radians
Sketch an arc of a unit circle measured in
standard position on the unit circle when:
5
x
3
*Note:
First, find the measure
of the corresponding
central angle for the
given radian measure.
HOMEWORK:
p. 717
#1-23 odd