Upload
raniksr
View
216
Download
0
Embed Size (px)
Citation preview
8/3/2019 (eBook) [Math] Trigonometry_rrr
1/23
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
8/3/2019 (eBook) [Math] Trigonometry_rrr
2/23
Maths Extension 1 Trigonometry
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
8/3/2019 (eBook) [Math] Trigonometry_rrr
3/23
Maths Extension 1 Trigonometry
3
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
8/3/2019 (eBook) [Math] Trigonometry_rrr
4/23
Maths Extension 1 Trigonometry
4
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
8/3/2019 (eBook) [Math] Trigonometry_rrr
5/23
Maths Extension 1 Trigonometry
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
8/3/2019 (eBook) [Math] Trigonometry_rrr
6/23
Maths Extension 1 Trigonometry
6
Area of a TriangleCabA sin
2
1! C is the angle a & b are the two adjacent sides
C
b a
8/3/2019 (eBook) [Math] Trigonometry_rrr
7/23
Maths Extension 1 Trigonometry
7
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
8/3/2019 (eBook) [Math] Trigonometry_rrr
8/23
Maths Extension 1 Trigonometry
8
1sin3sin2 2 UU = 0 1sin1sin2 UU = 0
Usin =2
1 Usin = 1
U = 210,330U = 270
8/3/2019 (eBook) [Math] Trigonometry_rrr
9/23
Maths Extension 1 Trigonometry
9
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
8/3/2019 (eBook) [Math] Trigonometry_rrr
10/23
Maths Extension 1 Trigonometry
10
= 1cos22
2 U
Utan =2
2
2
tan21
tan2U
U
8/3/2019 (eBook) [Math] Trigonometry_rrr
11/23
Maths Extension 1 Trigonometry
11
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
8/3/2019 (eBook) [Math] Trigonometry_rrr
12/23
Maths Extension 1 Trigonometry
12
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
8/3/2019 (eBook) [Math] Trigonometry_rrr
13/23
Maths Extension 1 Trigonometry
13
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
8/3/2019 (eBook) [Math] Trigonometry_rrr
14/23
Maths Extension 1 Trigonometry
14
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
8/3/2019 (eBook) [Math] Trigonometry_rrr
15/23
Maths Extension 1 Trigonometry
15
Area ofSegment
A = UU sin22
1 r
r
Segment
8/3/2019 (eBook) [Math] Trigonometry_rrr
16/23
Maths Extension 1 Trigonometry
16
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
8/3/2019 (eBook) [Math] Trigonometry_rrr
17/23
Maths Extension 1 Trigonometry
17
8/3/2019 (eBook) [Math] Trigonometry_rrr
18/23
Maths Extension 1 Trigonometry
18
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
8/3/2019 (eBook) [Math] Trigonometry_rrr
19/23
Maths Extension 1 Trigonometry
19
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
8/3/2019 (eBook) [Math] Trigonometry_rrr
20/23
Maths Extension 1 Trigonometry
20
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
8/3/2019 (eBook) [Math] Trigonometry_rrr
21/23
Maths Extension 1 Trigonometry
21
22
TTee y
xx 11 tan)(tan !
8/3/2019 (eBook) [Math] Trigonometry_rrr
22/23
Maths Extension 1 Trigonometry
22
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
8/3/2019 (eBook) [Math] Trigonometry_rrr
23/23
Maths Extension 1 Trigonometry
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