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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()± = 1tanθtβan 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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