Precalculus Section 4.1 Angles and Their Measures
Part 1: http://www.schooltube.com/video/887e91ffd00447feb411
 There are 360 degrees (360 ) in a circle. However, degrees can be subdivided.
 There are 60 minutes ( 60 ) in one degree, and there are 60 seconds  60 in one
minute.

Each minute is
1
1
th of a degree and each second is
th of a degree.
60
3600
EX) Convert 42 24 36 to a decimal value in degrees.
42 24 36 = ________________________________
EX) Convert 37.425
to DMS (Degrees, Minutes, and Seconds).
Step 1) ______ + _______
Step 2) ______ + _______
 60 

 =
 1 
37 o
+ ________
60
Step 3) 37 o + 25 + ______   = ______ + _______ + _______ = _____________
 1 
Degrees and Radians
 In trigonometry, we often measure angles with a new unit called the ____________.
 Radians can be used instead of ___________, but we need to define radians and
discover a way to convert degrees to radians.
Definitions
 A central angle of a circle has a measure of _____________if it intercepts an arc
with the same length as the ______________. (review central angles and intercepted arcs)
Turn to page 351, and see Figure 4.3.
a
a
1 radian
 Standard position of an angle: the vertex of the angle is at the__________, and the
initial ray is on the positive____________.
Part 2: http://www.schooltube.com/video/f363a049a3c243ad83dd
 A positive angle is rotated _______________________from standard position. The
terminal ray may rest in any quadrant.
y
x
 A negative angle is rotated __________________ from standard position. The
terminal ray may rest in any quadrant.
y
x
 Angles may be rotated (spun) more than 180 o in the positive or negative direction.
 Recall: Circumference of a circle: ____________
 Recall: All circles are similar.
If we consider a unit circle (circle of radius 1), then:
C  2 r becomes
C = _________
C = __________
 A revolution of 360 traces one _______________of a circle.
 Circumference is a _______________.

360 = ________ can be used to convert degrees to distance units called radians.
So, 360 degrees = _______ radians
OR 180
= _______ radians.
Part 3: http://www.schooltube.com/video/01036f5a0e0841f1a47f
Converting Degrees to Radians
degrees ___________  radians
EX) Convert the following degree measures to radians in terms of  and simplify.
a) 30
30
b) 60

180

30 


180
6
radians
60

180

60 


180
3
radians
c) 45
d) 270
e) 90
f) 360
g) 450
h) 180
Converting Radians to Degrees
radians
___________ = degrees
EX) Convert the following angle radian measures to degrees and simplify.
a)

3
 180
3 
c)

4
b)
radians


radians
6
 180
180

 30
6 
6
180
 60
3
d)
5
2
e) 3
Part 4: http://www.schooltube.com/video/5c65400e49ae476ca167
f)
5
6
g)
4
3
h)

2
Coterminal and Reference angles.
 Coterminal angles are angles that share the same ______________ ray. They may
be found by adding or subtracting multiples of full revolutions to or from a given
angle.
EX) 45o is coterminal with 315o , because if we add _______ (one full revolution) to 45o ,
we end up with a coterminal angle of 315o .
y
x
45
o
+ __________ = __________

11
radians is coterminal with 
radians, because if we add _______ (one full
6
6
11

revolution) to 
, we end up with a coterminal angle of
radians.
6
6
EX)
y
11
 2
6

11 12

6
6


6
EX) Find a negative angle coterminal with

3
x

.
3
 __________ = _______________ = _______________ which is coterminal with

.
3
Part 5: http://www.schooltube.com/video/6d1ba061373e43c1a1a5
Reference Angles
 Reference angle: A ____________ ___________angle formed between the terminal
ray of an angle and the ___________. (Shortest path to the x-axis)
 Reference angles are always ______________.
Let  = a given angle, and  = the reference angle for  .
y

y


x
x

EX) Find the reference angles for the following given angles.
Let the given angle be  and the reference angle be  .
a)  = 60 o
 = _______ because the shortest path to the x-axis from 60 o is _____.
b)  = 120o
 =_______ because the shortest path to the x-axis from 120o is _____.
c)  = 225o
 =_______ because the shortest path to the x-axis from 225o is ____.
d)  =
11
6
e)  = 
2
3
f)  = 190o
 = _______ because the shortest path to the x-axis from
11
is_____.
6
 =_______ because the shortest path to the x-axis from 
2
is_____.
3
 =________because the shortest path to the x-axis from 190o is______.
Part 6: http://www.schooltube.com/video/555718b44c1a43c6b93f
r
Circular Arc Length
sector -

s
 Because a central angle of 1 radian always intercepts an arc of _____________in
length, it follows that a central angle of  radians in a circle of radius r intercepts an
arc of length  r .
 Arc Length Formula (Radian Measure): If  is a central angle in a circle of radius r, and
if  is measured in radians, then the length s of the intercepted arc is given by:
s = _______________
 Arc Length Formula (Degree Measure): If  is a central angle in a circle of radius r, and
if  is measured in degrees, then the length s of the intercepted arc is given by:
s = ________________
EX) Find the arc length of a sector whose central angle is 30o and whose radius is 48 cm.
s
 r
180o
cm = ______________ = ______________ = ______________
EX) Find the perimeter and area of a sector whose central angle is

radians and whose
4
radius is 14 yards.
***Sector perimeter is: radius + radius + arc length.
Perimeter = _______ + _______ + _______ = _______ + ________
= _________ + _________ = ____________  _____________
Since

radians is ________ of a full revolution,
4
Sector Area is ____________ = ____________ = ____________  ________________