This document defines radians as a unit of measuring angles, where the radian measure of an angle is defined as the arc length of a unit circle subtended by the angle divided by the radius. It discusses converting between degree and radian measures, defines quadrantal angles in radians, and introduces the concept of coterminal angles which have the same terminal side.
22. Try ThisIf a 143°
angle is in
standard
position,
determine
the quadrant
in which the
terminal side
lies.
x
y
2
23. Try ThisIf a 280°
angle is in
standard
position,
determine
the quadrant
in which the
terminal side
lies.
x
y
4
24. Important Idea
There are two units of
measure for angles:
•degrees: used in geometry
•radians: used in calculus
In Precal, we use degrees
and radians.
25. -1 1
-1
1
y
x
Radian: The
length of the arc
above the angle
divided by the
radius of the
circle.
Definition
sr
θ
s
r
θ = , θ in radians (rads)
27. Definition
The radian measure of an
angle is the distance traveled
around the unit circle. Since
circumference of a circle is
2 r and r=1, the distance
around the unit circle is 2
π
π
28. Example
Find the degree
and radian
measure of the
angle in standard
position formed by
rotating the terminal side ½
of a circle in the positive
direction. Leave your radian
answer in terms of .π
29. Example
Find the degree
and radian
measure of the
angle in standard
position formed by
rotating the terminal side 5/6
of a circle in the negative
direction. Leave your radian
answer in terms of .π
30. Try This
Find the degree
and radian
measure of the
angle in standard
position formed by
rotating the terminal side 2/3
of a circle in the positive
direction. Leave your radian
answer in terms of .π
38. Important Idea
Radian measure allows the
expansion of trig functions
to model real-world
phenomena where
independent variables
represent distance or time
and not just an angle
measure in degrees.
39. Important Idea
If a circle contains 360° or 2π
radians, how many radians
are in 180°
• Use to change
rads to degrees
180°
π rads
• Use to change
degrees to rads
π rads
180°
53. Important Idea
To find coterminal angles,
simply add or subtract
either 360° or 2 radians
to the given angle or any
angle that is already
coterminal to the given
angle.
π
54. Analysis
30° and 390°
have the
same
terminal
side,
therefore,
the angles
are
coterminal
30°
x
y
x
y
390°
55. Analysis
30° and 750°
have the
same
terminal
side,
therefore,
the angles
are
coterminal
30°
x
y
x
y
750°