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Reference Guide

# Permachart - Marketing Reference Guide: Cartesian Coordinate System, Hypotenuse, S4N

4 Pages
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Fall 2015

Department
MATH - Math
Course Code
MATH 141
Professor
All
Chapter
Permachart

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Contributor: C. Bello, M.A.Sc.
ISBN: 1-55080-761-7
FUNDAMENTA L IDENTITIES
sin csc cos sec
tan cot
θθθθ
θθ
==
=
11
1
sin tan
sec cos cot
csc
tan sin
cos
θθ
θθθ
θ
θθ
θ
==
=
sin cos
tan sec
cot csc
22
22
22
1
1
1
θθ
θθ
θθ
+=
+=
+=
sin sin cos cos sin cos cos cos sin sin
tan tan tan
tan tan cot cot cot
cot cot
sin sin sin sin cos cos
cos cos cos sin
θβ θ β θ β θβ θ β θ β
θβ θβ
θβ θβ βθ
βθ
θβ θβ θ β β θ
θβ θβ θ
±
()
=± ±
()
=
±
()
=±±
()
=±
+
()
()
=−= −
+
()
()
=−
1
1
22 2 2
22222
ββθ
=−cos sin
sin sin cos tan
tan
cos cos sin sin
tan tan
tan
22 2
1
212
22
1
2
22 2
2
θθθ θ
θ
θθθ θ
θθ
θ
==
+
=−=
=
ri
nrnnin
rab n
cos sin cos sin
θθ θ θ
+
()
[]
=+
()
+
, where
= and = rational number
22
sin sin sin sin sin cos sin cos
cos cos cos cos cos cos
tan tan tan
tan tan tan tan
tan tan
sin sin cos sin
33 4 44 8
34 3 48 8 1
33
13 444
16
21
33
342
3
2
3
24
θθ θ θθθ θθ
θθθ θθθ
θθθ
θθθθ
θθ
θθθ
=− = −
=− =+
=
=
−+
=−
()
[]
−−nnn22
21 2 1
11
()
[]
=−
()
[]
−−
()
[]
=
()
[]
+
−−
()
[]
θ
θθθθθ
θθ
θθ
cos cos cos cos tan tan tan
tan tan
nn n n
n
n
sin sin sin cos
cos cos cos cos
cos cos sin sin
sin sin
sin sin
tan
tan
sin sin
cos cos tan
θβ θβ θβ
θβ θβ θβ
θβ θβ θβ
θβ
θβ
θβ
θβ
θβ
θβ θβ
±= ±
() ()
+= +
()
()
−=− +
()
()
+
=+
()
()
±
+
2
2
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
(()
±= ±
()
±=±
()
tan tan sin
cos cos cot cot sin
sin sin
θβθβ
θβ θββθ
θβ
sin sin cos cos sin cos sin sin
cos cos cos cos cos sin sin sin
θβ θβ θβ θ β θβ θβ
θβ θβ θβ θβ θβ θβ
=−
()
−+
()
=+
()
+−
()
=−
()
++
()
=+
()
−−
()
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
sin cos cos cos
tan cos
cos cot cos
cos
sin sin sin cos cos cos
21
2
21
2
22
31
4
31
4
12 12
12
12
12
12
33 3 3
θθ θθ
θθ
θθθ
θ
θθθ θ θθ
=−
()
=+
()
=
+=+
=−
()
=+
()
ei
ee
i
ee
iee
ee
ie
e
i
i
ii ii
ii
ii
i
i
θ
θθ θθ
θθ
θθ
θ
θ
θθ
θθ
θ
θ
=+
==+
=−
+
=−
+
−−
cos sin
sin cos
tan
22
1
11
2
2 where = -
q
/2 determines the
algebraic sign of the right-hand portion
RECIPROCAL RELATIONS QUOTIENT RELATIONS PYTHAGOREAN RELATIONS
POWER RELATIONS
FUNCTION-SUM/DIFFERENCE RELATIONS
ANGLE-SUM DIFFERENCE RELATIONS
DOUBLE-ANGLE RELATIONS
MULTIPLE-ANGLE RELATIONS
HALF-ANGLE RELATIONS
EXPONENTIAL RELATIONS
FUNCTION-PRODUCT RELATIONS
DE MOIVRES THEOREM
TRIGONOMETRIC RE L ATI O N S H I P S
sin cos tan
tan cot
sec
sec csc
sin cos
tan
cot
cot sec
csc
csc
sin
sin
cos
cos tan cot sec
csc
sin
sin
cos
θθ
θ
θθ
θ
θθ
θθ θ
θ
θθθ
θ
θ
θ
θ
θθθθθ
θ
θ
θ
±− ±+ ±+
±−
±− ±+ ±+
±−
±−
±− ±− ±−
±−
±
1
1
1
1
11
11
11
11
1
1111
1
1
1
2
22
2
2
22
2
2
22
2
2
−±
±−
±− ±+ ±+
±−
±−
±+ ±+ ±−
cos tan cot
sec
csc
sin cos tan cot
cot sec csc
csc
sin cos
tan
tan cot sec
sec csc
22
2
2
22
2
2
22
2
11
1
1
1
1
111
1
11
1
111
θθθθθ
θθθθ
θθθ
θ
θθ
θ
θθθ
θθ
sin
q
cos
q
tan
q
cot
q
sec
q
csc
q
sin
q
cos
q
tan
q
cot
q
sec
q
csc
q
Trade names are the property of their respective owners. Mindsource
Technologies Inc. and its partners disclaim all liability for any damage,
however caused, which may result from the application or misapplication
of this information. Visit permacharts.com, or call 1.800.387.3626.
l e a r n r e f e r e n c e r e v i e w
TM
permacharts
9781550807615
09560701362 1
TRIGONOMETRY • A-761-74www.permacharts.com

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Description
permachartsM l e a r n • r e f e r e n c e • r e v i e w TRIGONOMETRIC RELATIONSHIPS sinq cosq tanq cotq secq cscq tanθ 1 ± sec θ 1 1 sinq sinθ θ ±−1 cos2 ±+ 1 tan θ θ ±+ 1 cot secθ θ csc 2 cosq ±−1 sin θ θ cos 1 cotθ 1 ± −csc θ 1 ±+ 1 tan θ 2 ±+ cot θ secθ cscθ 2 tanq sinθ ±−1 cos θ tanθ 1 ± sec θ 1 1 2 cosθ cotθ 2 ±−1 sin θ ± csc θ 1 2 ±−1 sin θ cosθ 1 1 2 cotq sinθ 2 tanθ cotθ 2 ± −sc θ 1 − ± 1 −cos θ ± sec θ 1 1 1 2 ±+ 1 cot θ cscθ secq 2 cosθ ±+ 1 tan θ cotθ secθ 2 ±−1 sin θ ± −sc θ 1 1 1 ±+ 1 tan θ 2 secθ cscq 2 ±+ 1 cot θ 2 cscθ sinθ ±−1 cos θ tanθ ± −ec θ 1 FUNDAMENTAL IDENTITIES HALF-ANGLE RELATIONS RECIPROCAL RELATIONS QUOTIENT RELATIONS PYTHAGOREAN RELATIONS 2 2 1 1 −cosθ 1 1+cosθ sinθ = = 1 1 cosθ sinθ = =anθ cosθ cotθ sin θθc=os 1 sin2θ =± 2 cos2θ =± 2 cscθ secθ secθ cscθ 1 sinθ 1+tan θ θ sec 1 1−cosθ tanθ = tanθ = tan2θ =± 1+cosθ cotθ cosθ 1+c=ot θ θ csc • Quadrant location of q/2 determines the algebraic sign of the right-hand portion DE MOIVRE’S THEOREM DOUBLE-ANGLE RELATIONS ri osθθ sin n θ= r+θcosnin sin , where si22θθ =sin cos 2tanθ FUNCTION-SUM/DIFFERENCE RELATIONS [ ]) ( ) 1+tan θ 2 2 2 sinθ±s=in θ ±βθin() () cos1  r ab 2n and = rational number 2 cos2θ 1 cθos θsin sin 2 2 2 tanθ cosθ + =s βθ+βθcos2() cos21()− tan2θ= 2 ANGLE-SUM DIFFERENCE RELATIONS 1 −tan θ cosθ−− = +βθβθ sin21() sin2()− 1 si()β± =θinβ cosθ βcos sin ± θβ cos() θ βcos cos sin sin sinθ βsin tan2()β+ sinθ βsin 1 sinθ βsin = tan1 θβ− cosθ βcos = tan2 () tanθ βtan cotβcθot 1 2() tan()± = 1tanθtβan cot()β± = cotβ θcot si()± si()βθ± tanθ βt=an cotθ βc=ot si()β+ θsi()− θ = sβn 2 −siβn2 θcos cos cosθcβos sinθβin cos()+ coβ()− θ cos 2 2 sin β βc−os sin POWER RELATIONS 2 1 2 1 MULTIPLE-ANGLE RELATIONS sin θ =2( ) 2cθos θ co1 2= 2 ( )cos 3 3 1 2cos θ 1+cos θ si33θθ −s4n sin θ θθ 4 4 sin − 8 sin cos sin cos tan θ= 2 cot θ= cos34 θ −co33 cos θ θ θ = −4 8 cos8 1os cos 1 2cos θ 1−cos θ sin θ θθ 3sθin θθsin 3 cos = + cos cos 3tanθ θtan3 4 4nθ θ tan3 4( ) 4 ( ) tan3θ = 2 tan4θ = 2 4 13− tan θ 16 +tan θ θtan EXPONENTIAL RELATIONS sinθθ =2 1i() n cos − sin() 2θ [] [ ] e i= +osθ θ sin tan ()−1θ θ tan cosnθn=θ2−os[] n co2sn−c−o[ ] tan = [] e e −θ− − i θi θ e e + 1 tan[]) θ θtan sinθ = θ cos = 2i 2 FUNCTION-PRODUCT RELATIONS e e −θ−  e2iθ−1 tanθ =−  iθi θ− =− i 2iθ  whr= - i 1 sinθβin = θ−βco() θβ − +cos() β θβ sin θcβos = +si() + −sin() e e +  e +1 2 2 2 2 and θ is in radians cosθcβos = −cos()β + +coθ() θβ θβos sin = +si() − − si() 2 2 2 2 Contributor: C. Bello, M.A.Sc. Technologies Inc. and its partners disclaim all liability for any damage, ISBN: 1-55080-761-7 of this information. Visit permacharts.com, or call 1.800.387.3626.tion Printed in Canada. 9781550807615 070 1 4 TRIGONOMETRY • A-761-7 w w w .permacharts.com © 1997-2013 Mindsource Technologies Inc. l e a r n • r e f e r e n c e • r e v i e w permacharts TM Trigonometry T RIGONOMETRIC FUNCTIONS FORMULAS & GRAPHS GENERAL ANGLES SINE See also page 2. • q is any angle in y = 2sinx standard position X 2 y = sinx (i.e., vertex at origin of P(x,y) y = 1.5sin 2x Cartesian coordinate system, initial side 1 coincident with positive r x-axis) y θ • P(x,y) is any point on –2π –3π/2 –π –π/2 π/2 π 3π/2 2π the terminal side of the angle x –1 0 Y sinq = y/r cosq = x/r –2 tanq = y/x cotq = x/y secq = r/x cscq = r/y COSINE Memory Aid: Sine = Opposite ∏ Hypotenuse • Cosine = Adjacent ∏ Hypotenuse • Tangent = Opposite ∏ Adjacent S • O • H • C • A • H • T • O • A 2 y = 1.5cos2x 1 ACUTE ANGLES y = cosx • A and B are acute angles (< 90°) A • C is a right angle (= 90°) –2π –3π/2 –π –π/2 π/2 π 3π/2 2π • A + B + C = 180°; equations also hold for angles B and C c –1 b sinA = a/c cosA = b/c C –2 tanA = a/b cotA = b/a B y = 2cosx secA = c/b cscA = c/a a OBLIQUE ANGLES A • A and B are acute angles (< 90°) PERIOD, AMPLITUDE, & PHASE SHIFT • C is an oblique (obtuse) angle (> 90°) c • A + B + C = 180° b PERIOD Note: For equations • Function f is periodic if there is a positive real number k such relating to oblique B C that, for x and x + k in domain of f, f(x + k) = f(x) angles, see OBLIQUE TRIANGLES a • Fundamental period of f is smallest number p such that, for (page 3). x Œ R, f(x + p) = f(x) • Sine function has period of 2π; f(x) = f(x + p) if |kp| = 2π • So, f:x Æ asin (kx + d) is periodic with period 2π/|k| ALGEBRAIC SIGN OF FUNCTIONS AMPLITUDE sinq cosq tanq cotq secq cscq I + + + + + + • When a periodic function has maximum value of M and [0,1] [1,0] [0,∞ ) ( ∞ ,0] [1,∞ ) ( ∞ ,1] minimum value of m, amplitude is defined as (M – m)/2 • For f:x Æ sinx, x Œ R, it is known that –1 ≤ sinx ≤ 1, so M = 1 and II + – – – – + m = –1 [1,0] [0,–1] (–∞ ,0] [0,–∞ ) (
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