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TRIGONOMETRIC FUNCTIONS
OF ANY ANGLE
Letθ beanyangleinstandardposition,andlet , denote
thecoordinatesofanypoint,excepttheorigin 0, 0 ,onthe
denotesthedistancefrom
terminalsideof . If
0, 0 to , thenthesixtrigonometricfunctionsof are
definedastheratios:
Section 7.4
Trigonometric Functions
of General Angles
sin
cos
tan
csc
sec
cot
providednodenominatorequals0.Ifadenominatorequals
0,thattrigonometricfunctionoftheangleθ isnotdefined.
TRIGONOMETRIC FUNCTIONS
OF QUADRANTAL ANGLES
θ
0°;
0
90°;
π/2
180°;
π
270°;
3π/2
sinθ
cos θ
tanθ
csc θ
secθ
cotθ
0
1
0
not
defined
1
not
defined
1
0
not
defined
1
not
defined
0
0
−1
0
not
defined
−1
not
defined
−1
0
not
defined
−1
not
defined
0
COTERMINAL ANGLES AND
TRIGONOMETRIC FUNCTIONS
Becausecoterminal angleshavethesame
terminalside,thevaluesofthesix
trigonometricfunctionsofcoterminal angles
areequal.
COTERMINAL ANGLES
Twoanglesinstandardpositionaresaidtobe
coterminal iftheyhavethesameterminalside
NOTE: Coterminal anglesareNOT equal,they
merelystopatthesameplace.
SIGNS OF THE TRIGONOMETRIC
FUNCTIONS
Sign
of
sinθ
cscθ
cosθ
secθ
tanθ
cotθ
I
TerminalSideinQuadrant
II
III
IV
positive
positive
negative
negative
positive
negative
negative
positive
positive
negative
positive
negative
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THE REFERENCE ANGLE
THEOREM
REFERENCE ANGLES
Letθ denoteananglethatliesinaquadrant.
Theacuteangleformedbytheterminalside
ofθ andeitherthepositive ‐axisorthe
negative ‐axisiscalledthereferenceangle
forθ.
ReferenceAngleTheorem:Ifθ isanangle,in
standardposition, thatliesinaquadrantandα is
itsreferenceangle,then
sin
sin cos
cos tan
tan
csc
csc sec
sec cot
cot
wherethe+or−signdependsonthequadrantin
whichθ lies.
FINDING THE VALUES OF THE
TRIGONOMETRIC FUNCTIONS OF ANY
ANGLE
• Iftheangle isaquadrantal angle,drawtheangle,
pickapointonitsterminalside,andapplythe
definitionofthetrigonometricfunctions.
• Iftheangle liesinaquadrant:
1. Findthereferenceangle of .
2. Findthevalueofthetrigonometricfunctionat
.
3. Adjustthesign(+or−)ofthevalueofthe
trigonometricfunctionsbasedonthe
quadrantinwhich lies.
2