Trigonometry

577 Pages·2008·37.2 MB·English

by  Charles P.(Charles P. McKeague) McKeague

Charles P. McKeague
Mark D. Turner
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Operations on Complex Numbers in Standard Form (8.1] Products and Quotients in Trigonometric Form [8.3]
IfZI = al + bli and Z2 = a2 .,. b2i are two complex numbers in standard form, If<I = rl(cos III + i sin III) and Z2 = r2(cos 112 + i sin liz) then
then the following definitions and operations apply.
ZIZZ rlrz[cos (III + flzl + i sin (II, + (2)] = rlrZ cis (01 + 112)
Addition
ZAl d+d rZeza l pCaratls ;+ a adzdl i+m a(gbiln a+r yb 2p)air ts. <2 !"!. [cos (III fl2) i sin (01 - IIz)J . cis (III - 112)
r2 '2
Subtraction
Zl - Zz (al a2l + (bl - bz)i Oe Moivre's Theorem [8.3]
Subtract real parts; subtract imaginary parts. Ifz = r(cos II + i sin 0) is a complex number in trigonometric form and nis an
Multiplication integer, then
ZlZ2 (ala2 - bl b2) + (a lb2 + alb))i z" = r'(eos nil i sin nil) = t" cis (nil)
In actual practice, simply multiply as you would multiply two binomials.
ConJugates Roots of a Complex Number [8.4]
The conjugate of a + bi is a ­ bi. Their product is the real number
aZ + bZ• Thc nth roots of the complex number
Division z r(eos II + i sin e) rcis II
Multiply the numerator and denominator of the quotient by the conjugate are given by )j
of the denominator. ( 0 0
Wk = rlln rc os nfI +3;6-0 k) + i sin ( InI + -36n0- k
Graphing Complex Numbers [8.2]
= rliH cis I o + -360- 0 k ')
The graph ofthe complex number z x + yi is the arrow (vector) that extends n n
from the origin to fbe point (x, y). where k = 0, 1,2..... II - 1.
y
x + yi Polar Coordinates [8.5]
I
II The ordered pair (r. 0) names the point that is r units from fbe origin along the
I terminal side of angle II in standard position. The coordinates r and II are said
IY
I to be fbe polar coordinates of the point they name.
I
y
I' • x
x
8)
4" ·'i-""'i.',lv ~x
Absolute Value of a Complex Number [8.2]
The absolute value (or modulus) of the complex number x + yi is the dis­
tance from the origin to fbe point (x, y). If this distance is denoted by r, fben
r Izi = x + yi i
Polar Coordinates and Rectangular Coordinates [8.5]
Argument of a Complex Number [8.21 To derive the relationship between polar coordinates and rectangular coordi­
nates, we consider a point P with rectangular coordinates (x, y) and polar
fTrhome a trhgeu pmoesintti voef x t-haex ciso tmo pthleex g nraupmh boefrz .z I =f thxe +ar gyui mise fnbte o sfmzaisl ldeesnt optoesdi tbivye f l.a tnhgelne coordinates (r, e). y
sin II = 2::. cos II x and tan fI = X r = V x" + v" and tan 0 = "xv. . (x, y)
r r x (r, 8)
x r cos fI and y = r sin II
Trigonometric Form of a Complex Number [8.2]
The complex number z = x + yi is written in trigonometric form when it is I' x) I • x
written as
z = r(cos fI + i sin fI ) = rcis fI
where r is the absolute value of z and fI is the argument of z.
Exact Values on the Unit Circle [3.3] Inverse Trigonometric Functions [4.7]
y Inverse Functio_n_______________M_e_a_n_in-'g~~~~__~___ _
y 8in-1 x or y = arcsin x X = sin y and -­ 121 ' oS YoS 121'
In words: y is the angle between -11'/2 and 1T/2, inclusive, whose sine is x
Y cos­ 1 x or y = arccos x x = cos y and 0 oS Y oS 1T
In words: y is the angie between 0 and 1T, inclusive, whose cosine
is x,
__ (1,0) y = tan-1 x or y == arctan x X tan y and "72T < y < 72T
~-- ------------~------------~~--~x
In words: y is the angle between -1T/2 and1T/2 whose tangent is x,
Inverse Sine Inverse Cosine Inverse Tangent
y = sin-1 x y = cos-1 x y tan-Ix
arcsin x arccos x = arctan x
y y y
(x, y) (cos 0, sin I/)
Radian Measure [3.2] -t----7I'"----+.... x
-t----+----+.... x
-I
B (in radians) = S 5 Domain: Domain: Domain:
r
-I x -I xoSl all real numbers
Range: Range: Range:
Converting Between Radians and Degrees [3.2]
r + Uniform Circular Motion [3.4,3.5]
1~0 A point on a circle of radius r moves a distance s on the circumference of tb
Multiply by
circle, in an amount of time t,
Degrees Radians Angular Velocity Arc Length
o
~ w= 5 = rO
LMUltiPIYbY 1!0
Area of a Sector
s Linear Velocity
~r2ll
s A 2 v
v=­
Reference Angles [3.1] t
v rw
The reference angle {J for any angle fI in standard position is the positive acute fI is measured in radians
angle between the terminal side of 0 and the x-axis,
y The Area of a Triangle [7.4]
C
1
S 2ab sin C
s
I
wheres= 2(a+b c)
s= a 2 sin B sin C
2 sin A
DEFINITIONS, GRAPHS, AND IMPORTA T FACTS

Definition I for Trigonometric Functions [1 .3] Graphs of the Trigonometric Functions [4.1]
. in Ii = ',r- C ' 8 = ­,r. y (Assume k js any integer)
y

x r

l"(bil= ­ tl = ­
r x 3
-"
x 2 2
rol = ­
y -;;t:=::;::::=!-L----x
x ·f?--+-41r--+-.....~ x
Definition II for Trigonometric Functions [2.11
-2 -2 2
, B
-3 -3
.. ( =- - ~.~=­a
21"",,,,,,,,·,,,"
-t = ­b.. --4 =­ Domain: All real numbe . D main: All real num rs
Range: - ] ~y-l Range: -I ~yS I
A =­ ­ I and Jdja em 10 B Amplitude: I mplitud :

b .~= -' .~ b C Period: 2.. P riod:

id opfJO'-ite B :Uro : = k... Zer .:

:!lid adJ.. nI iD A

Definition III for Trigonometric Functions [3.3]
=.': :0-:: .... ; = - ,. = un r
'="- 1=­ 1.01. .r .' .­. x
~ ,
f .j
=;
Special Triangles [1.1J Domain: .A.ll real numbers Domain: All real n tim be rs
x= ", - ";; x= ...
-
Range: All real numbc-r;, Ran"e: All real numbers
Amplilud : . 'ot defined Amplitude: . 'ot de..fiD~d
Period: ;r Pe.-od: 1r
ZeT : x = :r; Zero : x= , - krr
Asympl te .\ = - - or; A"~mptOlc : x= ...
"- --9(f \. y
.'.=~\
Trigonometric Functions of Special Angles [2.1] .. 3 i\.
..
ti 0 6 .15' =Tr. 60' = T 900 = 2' 3Jr 2
2 "
x x
'\ 0 '\, :.: \ 3 \, 4 _1° ~(\If 2"
2 2 -2 2 Itf('\"I -2 2 2
t I I I
, \ 2 II VO' -3 It II -3 II II
8 2 2 2
Domain: All real numbers Domain: All real numbers
ian 6 0 V3 undefined
.t -...:.-L 72T ' I k 7i
Graphing Sine and Cosine Curves [4.2, 4.3] Ran ge : y:S - I or y 2:: 1 Range: \' :s - I or y 2:: I
Amplitude: Not defined Amplitude: NOl defined
Th graph, f,' = A in (Bx + C) and y = A cos (Bx + C). where B > O. will Period: 2'iT Period: 2r.
have the following characteristics: Zeros: None Zeros: None ..
Amplitude = A Peri od = B2" Phase shl. fl = -8C Asymptotes: X ::= kit Asy mptoles: x = 2 +br
IDENTI.TIES AND FORMULAS

Basic Identities [5.1] Product to Sum Formulas [5.5]
I
Common Equivalent sin A cos B = 2 [sin (A + B) + sin (A - Bll
Basic Identities Forms
I
Reciprocal csc () =-I ­ S.i n () =-1­ cos A sin B = 2 [sin (A + B) - sin (A - B)l
sin () csc ()
sec () = - 1- () cos () =-1­ cos A cos B = 2I [cos (A + B) + cos (A - B)l
cos sec ()
1 1 I
cot() =-­ tan () =-­ sin A sin B = 2[cOS (A - B) - cos (A + B)l
tan () cot(}
sin ()
Ratio tane =-­ e Pythagorean Theorem [1.1)
cos 2 J

e
cos
cote =-­
sin ()
Pythagorean cos2 e+ sin2 e = 1 si nl e = 1 - cos2 e
o c
sine = ::<:::Yl - cos2 e A
cos2 () = 1 - sin2e The law of Sines [7.1]
cos (} = ::<::: YI -sinle c
J + tan2 () = sec" e
1 + corl e= csc1 () b a
Sum and Difference Formulas [5.2J Half-Angle Formulas [5.4]
A c B
sin (A + B) = sin A cos B + cos A sin B si.n -A == ± (G-fcoisZ A sin A sin B sin C
sin (A - B) = sin A cos B - cos A sin B 2 \ 2 a o c
cos (A + B) = cosA cos B - sin A sinB A I + cos A or. equivalently.
cos (A - B) = cos A cos B + sin A sin B cos '2 -+- - -2 ­ a 0 C
tan (A + B) = tan A -'- tan B A - cosA sin A
sin A sin B sin C
I - tan A tan B tan '2 sin A .L cos A

The law of Cosines [7.3]
tan (A _ B) = tan A - tan B
I ..,-tan A tan B C
Even/Odd Functions [4.1]
cos (-8) = cos 8 Even o a
Double-Angle Formulas [5.3J
sin (-8) = - sin 8}
Odd

sin 2A = 2 sin A cos A tan (-8) = -tan 8

cos 2A = cos' A - sin' A First form A c B
2 cos' A - I Second form a' = 0' + c' - 20c cos A
= I - 2 sinl A Third form Sum to Product Formulas [5.5] b' = a' + c' - 2((c cos B
cr +j3 cr-j3
tan 2A = 2 tan A sin cr + sin {3 = 2 sin ~ cos - 2­ c' = a2 + 0' - 2ao cos C
I - tan' A
or. equivalently.
sin cr - sin j3 = 2 cos ~cr - j3 sin ~cr-j3 b2+c2-a1
cos A
20c
Cofunction Theorem [2.1]
sin x = cos (900 - xl cos cr + cos j3 = 2 cos ~cr +{3 cos -cr2-{3­ cos B = a2 -c2((1c -b1
cos x = sin (900 - x) cr+j3 cr - j3 (12 + b2 ­ c2
tan x = cot (900 - x) cos cr - cos j3 = -2 sin - -2- sin ~ cos C = 2ao
1
CHAPTER THE SIX TRIGONOMETRIC FUNCTIONS 1

2
CHAPTER RIGHT TRIANGLE TRIGONOMETRY 52

3
CHAPTER RADIAN MEASURE 109

4
CHAPTER GRAPHING AND INVERSE FUNCTIONS 169

5
CHAPTER IDENTITIES AND FORMULAS 255

6
CHAPTER EQUATIONS 299

7
CHAPTER TRIANGLES 337

8
CHAPTER COMPLEX NUMBERS AND POLAR

COORDINATES 389

A
APPENDIX REVIEW OF FUNCTIONS 443

B
APPENDIX EXPONENTIAL AND LOGARITHMIC

FUNCTIONS 464

ANSWERS TO EXERCISES AND

CHAPTER TESTS A-1

INDEX 1-1
tan
1.1 ANGLES, DEGREES, AND SPECIAL TRIANGLES 2

sir 1.2 THE RECTANGULAR COORDINATE SYSTEM 14

1.3 DEFINITION I: TRIGONOMETRIC FUNCTIONS 26

em
1.4 INTRODUCTION TO IDENTITIES 33

tal 1.5 MORE ON IDENTITIES 40

SUMMARY 45

TEST 48

PROJECTS 50

co
si
ta
2.1 DEFINITION II: RIGHT TRIANGLE TRIGONOMETRY 53

2.2 CALCULATORS AND TRIGONOMETRIC FUNCTIONS
OF AN ACUTE ANGLE 61

s 2.3 SOLVING RIGHT TRIANGLES 70

2.4 APPLICATIONS 81

2.5 VECTORS: A GEOMETRIC APPROACH 93

SUMMARY 104

TEST 106

PROJECTS 108

1
3.1 REFERENCE ANGLE 110

3.2 RADIANS AND DEGREES 117

3.3 DEFINITION III: CIRCULAR FUNCTIONS 129

3.4 ARC LENGTH AND AREA OF A SECTOR 140

3.5 VELOCITIES 150

SUMMARY 162

TEST 164

PROJECTS 166
Contents
i:,:~."'~#. u'I"~i~".'' i'f":"i ,I~,'~';:i~;0.:I,'.~.·.~:J·~. .:.I,'~,.tJ,'l:.y_;0.·i·.I;·.·.·,~ I_.'iI_ill.il._.,&J._
.. .. .. ...
4.1 BASIC GRAPHS 170

4.2 AMPLITUDE, REFLECTION, AND PERIOD 184

4.3 VERTICAL TRANSLATION AND PHASE SHIFT 197

4.4 THE OTHER TRIGONOMETRIC FUNCTIONS 208

4.5 FINDING AN EQUATION FROM ITS GRAPH 218

4.6 GRAPHING COMBINATIONS OF FUNCTIONS 231

4.7 INVERSE TRIGONOMETRIC FUNCTIONS 237

SUMMARY 249

TEST 252

PROJECTS 253

~~~.~~_J IDENTITIES AND FORMULAS 255

;(;\/0;:'4"," ;p; • i ~ ~ = :;",Jlf, '3f ",<"••,,,4
5.1 PROVING IDENTITIES 256

5.2 SUM AND DIFFERENCE FORMULAS 265

5.3 DOUBLE-ANGLE FORMULAS 275

5.4 HALF-ANGLE FORMULAS 282

5.5 ADDITIONAL IDENTITIES 288

SUMMARY 294

TEST 296

PROJECTS 297

EQUATIONS 299

6.1 SOLVING TRIGONOMETRIC EQUATIONS 300

6.2 MORE ON TRIGONOMETRIC EQUATIONS 308

6.3 TRIGONOMETRIC EQUATIONS INVOLVING

MULTIPLE ANGLES 313

6.4 PARAMETRIC EQUATIONS AND FURTHER GRAPHING 322

SUMMARY 332

TEST 333

PROJECTS 334
Contents
7.1 THE LAW OF SINES 338

7.2 THE AMBIGUOUS CASE 348

7.3 THE LAW OF COSINES 356

7.4 THE AREA OF A TRIANGLE 364

7.5 VECTORS: AN ALGEBRAIC APPROACH 369

7.6 VECTORS: THE DOT PRODUCT 378

SUMMARY 383

TEST 385

PROJECTS 387

D
C(
s
8.1 COMPLEX NUMBERS 390

8.2 TRIGONOMETRIC FORM FOR COMPLEX NUMBERS 398

t,
8.3 PRODUCTS AND QUOTIENTS IN TRIGONOMETRIC FORM 404

8.4 ROOTS OF A COMPLEX NUMBER 410

8.5 POLAR COORDINATES 417

8.6 EQUATIONS IN POLAR COORDINATES AND THEIR GRAPHS 427

SUMMARY 437

TEST 440

PROJECTS 441

A
REVIEW OF FUNCTIONS 443

APPENDIX
A.1 INTRODUCTION TO FUNCTIONS 443

A.2 THE INVERSE OF A FUNCTION 454

B
EXPONENTIAL AND LOGARITHMIC

APPENDIX
FUNCTIONS 464

B.1 EXPONENTIAL FUNCTIONS 464

B.2 LOGARITHMS ARE EXPONENTS 474

B.3 PROPERTIES OF LOGARITHMS 482

B.4 COMMON LOGARITHMS AND NATURAL LOGARITHMS 487

B.5 EXPONENTIAL EQUATIONS AND CHANGE OF BASE 494

ANSWERS TO EXERCISES AND CHAPTER TESTS A-1

INDEX 1-1