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