Lecture 25: Section 4.1
Angles and Their Measure
Angle - initial side, terminal side, vertex
Standard position of an angle
Positive and negative angles
Coterminal angle
Central angle
Radians
Complementary and suppplementary angles
Degree measure and radian measure
Arc length, s
Area of a sector
Linear speed
Angular speed
L25 - 1
An angle is formed by rotating a ray around its end-point. The starting position of the ray is the initial
side of the angle, and the position after rotation is theterminal side of the angle. The endpoint of the rayis called the vertex.
An angle θ is said to be in standard position if itsvertex is in the origin and its initial side coincides withthe positive x-axis.
x
y
L25 - 2
TERMINALSIDE
VERTEX 9INITIALSIDE
TERMINALSIDE
INITIALSIDEVERTEX
dNOTYn IN STANDARDPOSITIONra
L IX
v
If the rotation is in the counterclockwise direction, theangle is positive; if the rotation is clockwise, the angleis negative.
Angles α and β are coterminal angles if they havethe same initial and terminal sides.
x
y
x
y
A central angle is an angle whose vertex is at thecenter of a circle.
L25 - 3
µALSIDE
POSITIVE9 ANGLE INEGATIVE ANGLEINITIAL SIDE
2
IB
p
9
Radian Measure
Def. One radian is a measure of the centralangle that intercepts an arc whose length is equal tothe radius. Algebraically, this means that
θ =s
r
where θ is measured in radians.
NOTE: For the angle θ = 1 revolution:
The length of the arc (circumference) s = 2πr
Therefore, θ =s
r=
2πr
r= 2π.
We have,1 revolution = 2π radians
L25 - 4
DpE ANGLE'm p fnEut5 ARC LENGTHr RADIUS
ysttEEIIIEEIIEEAD.us90 5 rI l
r 0 1RADIAN
CIRCUMFERENCE ItdZrr
1 REVOLUTION 21TRADIANS
NOTE: If 0 ≤ θ ≤ 2π, the standard position of theangle θ in the Cartesian coordinate system is shownbelow:
NOTE: Given an angle θ, the coterminal angles to θare
θ + 2nπ
where n is an integer.
ex. Find the angle with the smallest positive measure
that is coterminal with θ = −21π
4.
Checkpoint: Lecture 25, problem 1
L25 - 5
TIZ
HEHEIt
IT2
21T n 2h ITe
n IS ANINTEGER
L 2141 21T 214T 8 13 h 3 24ft211433 247 2441
2 214 211423 216,1 Iq
Def. Given two positive angles α and β,
1. α and β are complementary if
2. α and β are supplementary if
ex. Find the complement and supplement of the angle
θ =π
7.
Checkpoint: Lecture 25, problem 2
L25 - 6
D2 t B _Iz ORd
Iz AND 900 ARE THE SAME THING j t go
Lt B ITL tB toooo
COMPLEMENTARY SUPPLEMENTARYANGLES ANGLES
pµ 7 f
7
92 92
LET B ANGLE WE ARE LOOKING FOR
COMPLEMENT
http LI F Iz E FIT 2FI B p
SUPPLEMENT
to it D T If 7GI 6 B 6
Degree Measure
Another way to measure angles is in terms ofdegrees, denoted by ◦.
1 counterclockwise revolution = 360◦
NOTE: 1 revolution = 360◦ = 2π rad
=⇒ 180◦ = π rad
Therefore, we have
1◦ =π
180rad and 1 rad =
180◦
π
L25 - 7
yI
800 431 30
Too Fo at I E HYE1 RAD 314 1200 600
LI iz141350I
1801T 0,3600 21T
IAi
i81 41 10615163 2700 3
312
Conversions between Radians and Degrees
Radian Degree
ex. Convert each angle in degrees to radians.
1) 60◦
2) 150◦
ex. Convert each angle in radians to degrees.
1)π
6
2) −3π
4
Checkpoint: Lecture 25, problem 3
L25 - 8
THISFRACTION REDUCES TO 1
180x TT
XII1800
Foo 98ft
ITToooo
150 ITTooo IT
6
40F 1861 300
19 3 41805 1350
Arc Length
For a circle of radius r, a central angle θ intercepts anarc of length s given by
s = rθ
where θ is measured in radians.
ex. A circle has a radius of 6 inches. Find the lengthof the arc intercepted by a central angle of 120◦.
Checkpoint: Lecture 25, problem 4
L25 - 9
PERIMETERt
PARTIAL CIRCUMFERENCE AROUNDCIRCLE
EARLIER SpS ARC LENGTHF RADIUSANGLE INRADIANS
5 ro
WANT 0 IN RADIANSno0 1200
Foo61T
120 ITTooo 3
5 61231 1231T 4ITNCH
Area of a Sector
For a circle of radius r, the area A of a sector withcentral angle θ is given by
A =1
2r2θ
where θ is measured in radians.
ex. A sprinkler sprays water over a distance of 30 feetwhile rotating through an angle of 150◦. What area oflawn receives water?
Checkpoint: Lecture 25, problem 5
L25 - 10
A ItrAREAOF A SECTOR
try
WANT 0 IN RADIANS0 1500 toooo
150
3 6
A Eto IIA 30.30 51T
2 6
375tTSQ