ANGLES
Section 7.1
Angles and Their Measure
POSITIVE AND NEGATIVE
ANGLES
• Anangleformedbyacounterclockwise
rotationisconsideredtobeapositive
angle.
• Anangleformedbyaclockwise rotationis
consideredtobeanegativeangle.
MEASURE OF ANGLES
Anglesaremostcommonlymeasuredin
degrees.Onedegree(1°)is ofa
completerevolution.
Degreescanbesubdividedintominutes
(notatedby’)andseconds (notatedby”)
asfollows:
1°
60 1
Anangle isformedbyrotatingarayaboutits
endpoint.Theoriginalrayistheinitialside
oftheangleandthesecondrayisthe
terminalside oftheangle.Thecommon
endpointiscalledthevertex oftheangle.
ANGLES AND THE xy-PLANE
Anangledrawnonthe ‐planeisin
standardposition whenitsinitialsideis
onthepositive ‐axis.
CLASSIFICATION OF ANGLES
• 90° anglesarerightangles.
• 180° anglesarestraightangles.
• Anglesthathaveameasurebetween0° and
90° areacuteangles.
• Anglesthathaveameasurebetween90°
and180° areobtuseangles.
60′′
1
ANGLE CLASSIFICATION AND
THE xy-PLANE
ANGLES AND CIRCLES
• Iftheterminalsideofanangleinstandard
positionliesonacoordinateaxis,thenthe
angleiscalledaquadrantal angle.
• Iftheterminalsideofanangleinstandard
positiondoesnotlieonacoordinateaxis,
thentheangleisclassifiedbythequadrant
thatcontainstheterminalside.
Considerthecirclebelowwithtworadii.The
angleformediscalledthecentralangle.The
partofthecirclebetweenA andB isanarc of
thecircleandiswritten AB
THE RADIAN
RADIAN MEASURE
Anothercommonmeasureofanglesisthe
radian.
Oneradian isthemeasureofthecentral
anglesubtendedbyanarcoflengthr ona
circleofradiusr.Anotherwaytothink
aboutthisismeasuringthedistanceofthe
radiusalongtheedgeofacircle.

Wesaythat AB subtends theangleθ.
Givenanarcoflengths onacircleofradiusr,
themeasureofthecentralanglesubtendedby
radians.
thearcis
Radianmeasureissimplyhowmanyradiiyou
mustgoaroundthecircle’sedgetogetthe
angle.
Radiansare“unitless.”
CONVERSIONS
To convert from radians to degrees, multiply
°
.
by
Toconvertfromdegreestoradians,multiply
by °.
ARC LENGTH
Letr bethelengthoftheradiusofacircleand
θ bethenonnegativeradianmeasureofa
centralangleofthecircle.Thenthelengthof
thearcs thatsubtendsthecentralangleis
.
2
EXAMPLES
1. Findthelengthofthearcthatsubtendsan
angleof135° onacircleofradius15cm.
2. Supposeabucketonawellisbeingraisedand
the“drum”hasaradiusof5inches.How
manyincheswillthebucketraiseifthedrum
isrotatedthroughanangleof75°?
3. ForthewellinExample2,supposeittakes
100revolutionstoraisethebucketfromthe
bottomofthewell.Approximately,howdeep,
infeetandinches,isthewell?
3