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1.3 Trigonometric Functions of any Angle
xD Deft If 0 is an anglein standard position and
o ih thepoint Cxy is anypoint on the terminalside of O then thesix trigonometric functionsof 0 are defined asfollows
r y since t cscO IyO f cost Xp Seco _IX tano L Coto L
We know x2ty2 r2 X yso xFtyI r
Note for the unit circle r 1so we have X cos O and y inO
On a circle of radius r x rcosOand y rsino
Example Find the six trigonometricfunctions of 0 if 0 is in standardposition and C 3,1 is on the terminalside of Owe could draw a picture IdSino g 1
Ipo TopoTE
r to 3
FOOT2 5 5
SCO or r fiftycosO If Zo go.ro
F3Ttf
3o9Mt Fo
v 10
seeO to3
tan O 9 tzX
Coto 3
The first quadrant of the unit circleQuadrantalAngles co The x value is
1 the cosine of thegoetz angle and the
y value is theC id 1800 51 o cb0 sine of theangleI I
2700
Iz e.g Cos OEb 1 sin 1800 0
Oi Edcos 1800 1
First Quadrantcry r
sin 321 1
ez d tan 0 9 0i am coto.IE
aedLSeciT
circle ofradius2 II2 6h00 I I
307 f 30 TZ
B B2
Simple trick to help you recall 1stquadrantRadius O coff Sino0 00 l r O Roz Pattern tothing
If 300 Be rt cosy're sione
If 450 E Ez f fIf 600 E L Bz O 4
IL 900 E O l
z 2
Examples E
sin Iiicos If Fez
tan Ee Eze F E toFz Erg FF
see 600 cost Iz 2
We can get the other three quadrantsusing symmetry but we will do thata bit laterReferenceAngles
8 is the angle we formi To when we use aperpendicular8 i to the X axis to form
a triangle
bet the reference angledenoted 8 for an angle0 in standardposition isthe positive acute angleCcosoSino
4 in between the terminalis side of O and the
x axis
Examples find the reference angle2ndquadrant0 1500180 1500 300 4
1800 I 8 I
3rdquadranta 2250I 225 180 1800 250 I
4504thquadrant
13 285 28613 360 285 iffy 28755750 Leafs
0 1200600I200 t 1800 600 Con i
aka6It Gon 7 I
8 413
Ig A 3If 0
HIT
015 IfI
21T 8
4 4
Note Coterminal angle1541.2 15g I47 24T 741
8 1
Quadrants is cosinepositivetwanffinutesingosinenegatiret I sinepfasngiteinfpositive
T t tt sincet
sinepositive L
cosine is sine is negativenegative C r tishe is Rosine is positivenegative
tangent is positive tangent is negativesince
t since I
Example Given since and0 terminates in quadrant II Findthe remaining trigonometric functionsof 0sin 0 12,3 Gosoisin
cos8 73 cos0 73TOseed secO13
2.4cg tano tano 12gX
Sino fIz oPP cot 8 fz cotO Esfoindwthe csc8 cscoEside it 144 169 sin tg sinO
144 1442 25X ffs 5
We could have found sin 8 andcos 8 to get sin and costand then used the identities toget the rest e.g tano going IIE
13
ifRecall the first quadrant of the
unit circle
8 O neo i E300 BE Iz Iz f E
450kloci
600 I I BZ Z
900 Ez O Ftl co 1
Examples o n choose
o d
cosy1500 I eonskoo I EE Ez
Find the reference INangle and choose 1800 00
sign based onquadrant
choose
sin 2250 I sin450 ETralaate 2
180y
0
270
tan C 4800 tan
2407gFind coterminal angle first for ease480 t 360 120120 t 360 2400
1800 o
II I2700
I tan 600 I sin 600 Bz
y IsaoI t
If B
sin901Csc 4500 Csc 900 coil
Cotermind 450 singoo T Lgo
36900
Sec 4500 Secc900
cost toundefined
If we do not have a referenceangle of 300 450 or 600 or one
of the quadrantal angles then we mayneed a calculator goo
seal 570 o1780 1800 0015723
I see 230 230
2I 1.09