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Reference Guide

Trigonometry - Reference Guides

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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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