Section 4.1: Angles and Their Measures
Homework: 2, 3, 8, 9, 14, 15, 18, 23, 26, 31, 34, 38,
39,42,43,45,47,49,54,59,61,66,67,72
UWarm-up
How many degrees does the minute hand of a clock travel in 45 minutes? 50 minutes?
I ~ mit)~qoo
mIn --} \ro()
?fJ
45
~ 210
m '\r\
0
45 + 5
-4
210
0
+ 5(~/')
-=-
300
0
UDegrees & Radians
•
DIVIS (Degree-Minute-Second) system of angular measure (used for accuracy)
•
Each degree is subdivided into 60 minutes.
•
Each minute is subdivided into 60 seconds .
De3ree ~ decnna l form
~DMS
Example:
(a) Convert
37.42S~to DMS
(t=t5 l1l-l)
(b) Convert 42° 24' 36" to degrees .125(~~ )-::25 .5' 5
(b /O )
30
-::
12" t (~~r + (3it
\1
12.11
31° 25\ QO\I
0
In Exercises 1-4, convert from DMS to decimal form.
2.35°24'
4. 48°30 /36//
(!) 23°12' ®
118°44 ' 15// In Exercises 5-8, convert from decimal form to degrees, minutes,
seconds (DMS).
5) .2 (1010)
-=:
@ 21.2° 6.49 .7"
® 1l8.32° 8. 99.3r
12 \
'1) . 62
(~O)
-2 (~) ~
21 0 12 \ 118
0
19
1
1211
7:
IQ , 2
1/
r
I) 23
0
t-
(g)O
loo .
= 23 .2°
3) ii\)' + (~ ) t (~
=liS .1315 0
Rod IQf\S ".
Degree-Radian Conversion
. 1Y b Y
To convel1 radians to degrees, rou 1up
0
180
di
'Tl'ra ans
·
. I Y bY 'Tl'radians
'Iio convert degrees to radlans,
mu Itip
180 0 •
LeqVe In
terms of 'iT
or iabel
II
rQdlOns II
OS
~~~~~~b~e~~~~~~~~~I
. ~________________________~
In Exercises 17-24, convert from radians to degrees.
In Exercises 9-16, convert from DMS to radians.
10.90°
18.1T/4
12. 150°
@
1T/5
22. 131T/20
14.11.83°
@
1\)
120-* 1T
-
120TI
::-
24.1.3
21T
3
-:::
ieD
180
/3) ·ll .lZ +-rr
IBo
5°30'
1112n ':: .3q B'tl
180 ~ /25 rad
-::
"Circular Arc Length Arc Length Formula (Radian Measure) S
e
If is a central angle in a circle of radius r, and
the length s of the intercepted arc is given by
il eis measured in radiansl then
s = reo
r
s
r
1.5ft
?
17/ 4 rad
28.2.5 em
?
'Tl'/3 rad
29.3 m
1m
')
7 in.
?
@
@
4in.
?
5ft
Use the appropriate arc length formula to
find the missing information.
2l)
J.
5 ~ r (TT 11 )
Example : Find the perimeter of a 60° slice of pizza from a 7" radius pizza. s~re
r-= '
7
s
e~
'
£00
0
"'
\1
1"
Example:
1
p-= 1 1 t 1,33 = 1\. 33
-TrI~ _- IT3
i
~ (hav'n~
a
s~l"'TT'!-l . ~'
C£NlW)
common
3
The concentric circles on an archery target are 6 inches apart. The inner circle (red) has
perimeter of 37.7 inches. What is the perimeter of the next-largest (yellow) circle?
Perlmeter-4 Circumference
6"
~e \lOLD
Red
31 .1 ~ LTT r
r-= 31 1
~
t~\I
r-= . b'l
(lrr )
r':. fa
G= 2rrr
it
)
154­
("'2fT (
J2
c= 2Yrr
~
j 2\\
~
In·
Example:
The running lanes at the Emely Sears track at Bluffton College are 1 meter wide. The
inside radius of lane 1 is 33 meters and the inside radius of lane 2 is 34 meters . How
much longer is lane 2 than lane 1 around one tum ? (See Figure 4.5.)
s:re
Lane l
«
34m
~
S'"
e"
re
I~O° =- IT
r ~ -33
s-= ~~rr
@:
\\)3 ·1 ~
lone 2
S-:or8
8 iOO°
~-rr
-:0
r
-'=
34­
s~
~ IO~ -~
HOw mu(h lGnger ~ 106 .e -\6'6.1 ~
~.
i
**Angular & Linear Motion
Example:
Cathy races on a bicycle with 12" radius wheels. When she is traveling 44 ft/sec, how many
revolutions per minute (rpm) are her wheels making?
rev
12 jt1'
2411%
Iff
I
44 ft. I SOC i -t 11 ... 11 .... ~ rev ~ 31 fa <00 -=
24rr -t ) t I * J mlr)
2~ IT
410
Example:
In navigation, the bearing of an object is sometimes given as the angle of the line of travel measured
clockwise from due north.
A
then
pi \0 t travel $ 0:>
changec io
Q
bearl nj of 33
80° for
10 miles.
0
for £J Draw .
rpm