(ebook) [math] trigonometry_rrr

8/3/2019 (eBook) [Math] Trigonometry_rrr

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Maths Extension 1 Trigonometry

1

Trigonometry

Trigonometric RatiosExact Values & TrianglesTrigonometric IdentitiesAS

TC RuleTrigonometric GraphsSine & Cosine RulesArea of a TriangleTrigonometric EquationsSums and Differences of anglesDouble AnglesTriple AnglesHalfAngles

T formulaSubsidiary Angle formulaGeneral Solutions of Trigonometric EquationsRadiansArcs, Sectors, SegmentsTrigonometric LimitsDifferentiation of Trigonometric FunctionsIntegration of Trigonometric FunctionsIntegration of sin

2x and cos

2x

INVERSE TRIGNOMETRYInverse Sin Graph, Domain, Range, PropertiesInverse Cos Graph, Domain, Range, PropertiesInverse Tan Graph, Domain, Range, PropertiesDifferentiation of Inverse Trigonometric FunctionsIntegration of Inverse Trigonometric Functions


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2

Trigonometric Ratios

Sine sinU =hypotenuse

opposite

Cosine cosU =hypotenuse

adjacent

Tangent tanU =adjacent

opposite

Cosecant cosecU =Usin

1 =

opposite

hypotenuse

Secant secU =Ucos

1 =

adjacent

hypotenuse

Cotangent cotU = Utan

1

=

opposite

adjacent

sinU = Ur90cos cosU = Ur90sin tanU = Ur90cot cosecU = Ur90sec secU = Ur90cosec cotU = Ur90tan

60 seconds = 1 minute 60 = 1

60 minutes = 1 degree 60 = 1

UU

Ucos

sintan !

UU

Usin

coscot !

hypotenusehypotenuse

opposite

adjacent

adjacent

oppo

site


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Exact Values & Triangles

0 30 60 45 90 180sin 0 2

1 2

3 2

1 1 0

cos 1 23 2

1 2

1 0 1

tan 0 31 3 1 0

cos ec 2 32 2 1

sec 1 32 2 2 1

cot 3 31 1 0

Trigonometric Identities

UU 22 cossin = 1

U2cos = U2sin1

U2sin = U2cos1

U2cot1 = cosec2U

U2cot = cosec2

U 1

1 = cosec2U U2cot

1tan2 U = U2sec

U2tan = 1sec2 U 1 = UU 22 tansec

1

1

2

45

3

1

2

30

60


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ASTC Rule

First Quadrant: All positive

Usin Usin +

Ucos Ucos +

Utan Utan +

Second Quadrant: Sine positive Ur180sin Usin + Ur180cos Ucos

Ur180tan Utan

Third Quadrant: Tangent positive Ur180sin Usin Ur180cos Ucos Ur180tan Utan +

Fourth Quadrant: Cosine positive Ur360sin Usin Ur360cos Ucos + Ur360tan Utan

rr

3600

90

180

270

S A

T C

1st

Quadrant

4th

Quadrant

2nd

Quadrant

3rd

Quadrant


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5

Trigonometric Graphs

Sine & Cosine Rules

Sine Rule:

C

c

B

b

A

a

sinsinsin!! OR

c

C

b

B

a

A sinsinsin!!

Cosine Rule:

Abccba cos2222 !

A

BC

a

b c

A

a

b c


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Area of a TriangleCabA sin

2

1! C is the angle a & b are the two adjacent sides

C

b a


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Trigonometric Equations

Check the domain eg. reer 3600 U Check degrees ( reer 3600 U ) or radians ( TU 20 ee )If double angle, go 2 revolutionsIf triple angle, go 3 revolutions, etcIf half angles, go half or one revolution (safe side)

Example 1Solve sin =

2

1 for reer 3600 U

Usin =2

1

U = 30, 150

Example 2

Solve cos 2 = 21 for reer 3600 U U2cos =

2

1

U2 = 60, 300, 420, 660

U = 30, 150, 210, 330

Example 3

Solve tan2

U = 1 for reer 3600 U

tan2

U = 1

2

U

= 45, 225U = 90

Example 4

0cos2sin ! UU

UUU coscossin2 = 0 1sin2cos UU = 0

Ucos = 0 Usin = 21

U = 90,270

U = 210,330

Example 5

22cossin3 ! UU UU 2sin21sin3 = 2


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1sin3sin2 2 UU = 0 1sin1sin2 UU = 0

Usin =2

1 Usin = 1

U = 210,330U = 270


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Sums and Differences of angles FE sin = FEFE sincoscossin FE sin = FEFE sincoscossin FE cos = FEFE sinsincoscos

FE cos = FEFE sinsincoscos FE tan =

FEFE

tantan1

tantan

FE tan =FEFE

tantan1

tantan

Double Angles

U2sin = UU cossin2

U2cos = UU 22 sincos = U2sin21

= 1cos2 2 U

U2tan =U

U2tan21

tan2

U2sin = U2cos12

1

U2cos = U2cos121

Triple Angles

U3sin = UU 3sin4sin3

U3cos = UU cos3cos4 3

U3tan =U

UU2

3

tan31

tantan3

HalfAngles

Usin =22

cossin2 UU

Ucos =2

2

2

2 sincos UU

=2

2sin21 U


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10

= 1cos22

2 U

Utan =2

2

2

tan21

tan2U

U


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Deriving the Triple Angles

U3sin = UU 2sin = UUUU sin2coscos2sin

= UUUUU sinsin21coscossin2 2

= UUUU32

sin2sincossin2 = UUUU 32 sin2sinsin1sin2 = UUUU 33 sin2sinsin2sin2

= UU 3sin4sin3

_Normal double angle_Expand double angle_

Multiply_Change

1cossin 22 ! UU _

Simplify_

U3cos = UU 2cos = UUUU sin2sincos2cos = UUUUU sincossin2cos1cos2 2 = UUUU cossin2coscos2

23

= UUUU coscos12coscos2 23 = UUUU 32 cos2cos2coscos2

= UU cos3cos4 3

U3tan = UU 2tan

=UUUU

tan2tan1

tan2tan

=U

UUU

U

U2

2

tan1

tantan2

tan1

tan2

1tan

=U

UUU

UUU

2

22

2

3

tan1

tan2tan1

tan1

tantantan2

=U

UU2

3

tan31

tantan3


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T Formulae

Let t = tan2

U

Usin = 212

t

t

Ucos = 22

1

1

t

t

Utan = 212

t

t

Usin =22

cossin2 UU

=2

2

2

2

22

sincos

cossin2UU

UU

=

2

2

2

2

2

2

22

22

cos

sincos

cos

cossin2

U

UU

U

UU

=2

2

2

tan1

tan2U

U

=21

2

t

t

Using half angles_

Divide by 1

1cossin 22 ! UU

Divide top and bottom by

U2cos

cos cancel;cos

sin becomes tan

Ucos =2

2

2

2 sincos UU

=2

2

2

2

2

2

2

2

sincos

sincosUU

UU

=

2

2

2

2

2

2

2

2

2

2

2

2

cos

sincos

cos

sincos

U

UU

U

UU

=2

2

2

2

tan1

tan1U

U

=2

2

1

1

t

t

Utan =UU

cos

sin

=2

2

2

1

1

1

2

t

t

t

t

=21

2

t

t


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Subsidiary Angle Formula

xbxa cossin = )sincoscos(sin xxxxR

= xxRxxR sincoscossin

a

= xRcos 2

a@

= xR22

cos b = xRsin 2b@ = xR 22 sin

1cossin 22 ! xx =2

22

R

ba

22 baR ! a

b!Etan

xbxa cossin = C )sin( ExR

xbxa cossin = C )sin( ExR

xbxa sincos = C )cos( ExR

xbxa sincos = C )cos( ExR

Example 1Find x. 1cossin3 ! xx

R = 22

13 Etan =3

1

= 4

= 2 E = 30

)30sin(2 x )30sin( x

30x

x

= 1

=2

1

= 30, 150= 60, 180


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General Solutions of Trigonometric Equations

EU sinsin ! Then ETU nn )1(!

EU coscos ! Then ETU s! n2

EU tantan ! Then ETU ! n

RadianscT = 180

1 =180

cT

Arcs, Sectors, Segments

Arc Length

l = Ur

Area ofSector

A = U22

1r

l

r

r


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Area ofSegment

A = UU sin22

1 r

r

Segment


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Trigonometric Limits

x

x

x

sinlim

0p

=x

x

x

tanlim

0p = x

x

coslim0p

= 1

Differentiation of Trigonometric Functions

xdx

dsin

= xcos

? A)(sin xfdx

d

= )(cos)(' xfxf

)sin( baxdx

d

= )cos( baxa

xdx

dcos

= xsin

? A)(cos xfdx

d

= )(sin)(' xfxf

)cos( baxdx

d

= )sin( baxa

xdxd tan

= x2sec

? A)(tan xfdx

d

= )(sec)(' 2 xfxf

)tan( baxdx

d

= )(sec2 baxa

xdx

dsec

= xx tan.sec

ecxdx

dcos

= ecxx cos.cot

xdx

dcot = xec2cos


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Integration of Trigonometric Functions

axcos dx = caxasin

1

axsin dx = caxa cos

1

ax2sec dx = cax

atan

1

22

1

xadx = c

a

x

1sin

22

1

xadx = c

a

x

1cos __OR__ ca

x

1sin

221

xadx = c

a

x

a

1tan1

axec2cos dx = cax

a cot

1

axax tan.sec dx = caxasec

1

axecax cot.cos dx = cecaxa cos

1


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Integration of sin2x and cos

2x

x2cos

12cos x 12cos

2

1 x

= 1cos2 2 x = x2cos2 = x2cos

x2

cos dx = 12cos2

1

x dx= Cxx 2sin

2

1

2

1

= Cxx 2

1

4

1 2sin

x2cos dx = Cxx

2

1

4

1 2sin

x2cos

x2sin2

x

2

sin

= x2sin1 = x2cos1

= x2cos12

1

x

2sin dx = x2cos121 dx

= Cxx 2sin2

1

2

1

= Cxx 2sin4

1

2

1

x2sin dx = Cxx 2sin

4

1

2

1


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INVERSE TRIGNOMETRY

Inverse Sin Graph, Domain, Range, Properties

11 ee x

22

TTee y

xx 11 sin)(sin !

Inverse Cos Graph, Domain, Range, Properties

11 ee x

Tee y0

xx 11 cos)(cos ! T

Inverse Tan Graph, Domain, Range, Properties

All real x

2

T

2

T

T

2

T

1-1

x

y

0

2-2

x

y

2

-2

x

y

2

T

2

T


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21

22

TTee y

xx 11 tan)(tan !


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Differentiation of Inverse Trigonometric Functions

xdx

d 1sin =21

1

x

a

x

dx

d 1sin

= 221

xa

)(sin 1 xfdx

d =2)]([1

)('

xf

xf

xdx

d 1cos =21

1

x

a

x

dx

d 1

cos

= 221

xa

)(cos 1 xfdx

d =2)]([1

)('

xf

xf

xdx

d 1tan =21

1

x

a

x

dx

d 1tan =22 xa

a

)(tan 1 xfdx

d =2)]([

)('

xfa

xf


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Integration of Inverse Trigonometric Functions

22

1

xadx = c

a

x

1sin

22

1

xadx = c

a

x

1cos __OR__ ca

x

1sin

221

xadx = c

a

x

a

1tan1

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