Trig Cheat Sheet Formulas And Identities

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Formulas and IdentitiesTrig Cheat SheetDefinition of the Trig FunctionsRight triangle definitionFor this definition we assume thatp0 q or 0 q 90 .2Unit circle definitionFor this definition q is any angle.y( x, y )hypotenuseyopposite1qxxqtan q 1 sec qadjacentoppositesin q hypotenuseadjacentcos q hypotenuseoppositetan q adjacentTangent and Cotangent Identitiessin qcos qtan q cot q cos qsin qReciprocal Identities11csc q sin q sin qcsc q11sec q cos q cos qsec q11cot q tan q tan qcot qPythagorean Identitiessin 2 q cos 2 q 12hypotenusecsc q oppositehypotenusesec q adjacentadjacentcot q oppositey y1xcos q x1ytan q xsin q 1y1sec q xxcot q ycsc q Facts and PropertiesDomainThe domain is all the values of q thatcan be plugged into the function.sin q , q can be any anglecos q , q can be any angle1ˆÊtan q , q π Á n p , n 0, 1, 2, 2 Ëcsc q , q π n p , n 0, 1, 2, 1ˆÊsec q , q π Á n p , n 0, 1, 2, 2 Ëcot q , q π n p , n 0, 1, 2, RangeThe range is all possible values to getout of the function.csc q 1 and csc q -1-1 sin q 1-1 cos q 1 sec q 1 and sec q -1- tan q - cot q PeriodThe period of a function is the number,T, such that f (q T ) f (q ) . So, if wis a fixed number and q is any angle wehave the following periods.2pw2p wp w2p w2p wp wsin ( wq ) ÆT cos (wq ) ÆTtan (wq ) ÆTcsc (wq ) ÆTsec (wq ) ÆTcot (wq ) ÆT 2005 Paul Dawkins21 cot 2 q csc 2 qEven/Odd Formulassin ( -q ) - sin qcsc ( -q ) - csc qcos ( -q ) cos qsec ( -q ) sec qtan ( -q ) - tan qcot ( -q ) - cot qPeriodic FormulasIf n is an integer.sin (q 2p n ) sin qcsc (q 2p n ) csc qcos (q 2p n ) cos q sec (q 2p n ) sec qtan (q p n ) tan qcot (q p n ) cot qDouble Angle Formulassin ( 2q ) 2sin q cos qcos ( 2q ) cos 2 q - sin 2 q 2 cos 2 q - 1 1 - 2sin 2 q2 tan qtan ( 2q ) 1 - tan 2 qDegrees to Radians FormulasIf x is an angle in degrees and t is anangle in radians thenptpx180t fi t and x 180 x180pHalf Angle Formulas1sin 2 q (1 - cos ( 2q ) )212cos q (1 cos ( 2q ) )21 - cos ( 2q )2tan q 1 cos ( 2q )Sum and Difference Formulassin (a b ) sin a cos b cos a sin bcos (a b ) cos a cos b m sin a sin btan a tan b1 m tan a tan bProduct to Sum Formulas1sin a sin b ÈÎcos (a - b ) - cos (a b ) 21cos a cos b ÎÈ cos (a - b ) cos (a b ) 21sin a cos b ÎÈsin (a b ) sin (a - b ) 21cos a sin b ÈÎsin (a b ) - sin (a - b ) 2tan (a b ) Sum to Product FormulasÊa b ˆÊa - b ˆsin a sin b 2 sin Á cos Á Ë 2 Ë 2 Êa b ˆ Êa - b ˆsin a - sin b 2 cos Á sin Á Ë 2 Ë 2 Êa b ˆÊa - b ˆcos a cos b 2 cos Á cos Á Ë 2 Ë 2 Êa bcos a - cos b -2sin ÁË 2Cofunction Formulasˆ Êa - b ˆ sin Á Ë 2 Êpˆsin Á - q cos qË2 Êpˆcsc Á - q sec qË2 Êpˆcos Á - q sin qË2 Êpˆsec Á - q csc qË2 Êpˆtan Á - q cot qË2 Êpˆcot Á - q tan qË2 2005 Paul Dawkins

Unit CircleyÊ3 1ˆÁ- , Ë 2 2 ( -1,0 )3p45p6( 0,1)p2Ê 1 3ˆÁ- , Ë 2 2 Ê2 2ˆ,Á Ë 2 2 Inverse Trig FunctionsDefinitiony sin -1 x is equivalent to x sin y2p3p390 120 y cos -1 x is equivalent to x cos yÊ1 3ˆÁÁ 2 , 2 Ë -145 135 y tan x is equivalent to x tan yÊ 2 2ˆ,ÁÁ Ë 2 2 p460 30 p6Domain and RangeFunctionDomainÊ 3 1ˆÁÁ 2 , 2 Ë y sin -1 x-1 x 1-1150 p 180 0 0360 2py cos x-1 x 1y tan -1 x- x (1,0 )Ê3 1ˆÁ - ,- Ë 2 2 7p6Ê22ˆ,Á 2 Ë 2330 225 5p44p3240 Ê 13ˆÁ - , Ë 2 2 315 7p300 270 45p3p32Ê( 0,-1)11p6For any ordered pair on the unit circle ( x, y ) : cos q x and sin q ycos -1 x arccos xtan -1 x arctan xbagLaw of Sinessin a sin b sin g abcLaw of Tangentsa - b tan 12 (a - b ) a b tan 12 (a b )Law of Cosinesa 2 b 2 c 2 - 2bc cos ab - c tan 12 ( b - g ) b c tan 12 ( b g )c a b - 2ab cos g2Ê 5p ˆ 1cos Á Ë 3 2tan -1 ( tan (q ) ) qbb 2 a 2 c 2 - 2ac cos bExampletan ( tan -1 ( x ) ) xaÊ 3 1ˆÁ ,- Ë 2 2 13ˆÁ , Ë2 2 sin -1 ( sin (q ) ) qLaw of Sines, Cosines and TangentsxÊ 22ˆ,Á 2 Ë 2sin ( sin -1 ( x ) ) xAlternate Notationsin -1 x arcsin xRangepp- y 220 y ppp- y 22c210 Inverse Propertiescos ( cos -1 ( x ) ) xcos -1 ( cos (q ) ) q22a - c tan 12 (a - g ) a c tan 12 (a g )Mollweide’s Formulaa b cos 12 (a - b ) csin 12 g3Ê 5p ˆsin Á 2Ë 3 2005 Paul Dawkins 2005 Paul Dawkins

Common Derivatives and IntegralsCommon Derivatives and IntegralsDerivativesIntegralsBasic Properties/Formulas/Rulesd( cf ( x ) ) cf ( x ) , c is any constant. ( f ( x ) g ( x ) ) f ( x ) g ( x )dxd nd( c ) 0 , c is any constant.( x ) nxn-1 , n is any number.dxdxÊ f ˆ f g - f g – (Quotient Rule)( f g ) f g f g – (Product Rule) Á g2Ëg df ( g ( x ) ) f ( g ( x ) ) g ( x ) (Chain Rule)dxg ( x)dd g ( x)g xln g ( x ) ) e g ( x ) e ( )(dxg ( x)dx()( )Common DerivativesPolynomialsdd(c) 0( x) 1dxdxd( cx ) cdxTrig Functionsd( sin x ) cos xdxd( sec x ) sec x tan xdxd( cos x ) - sin xdxd( csc x ) - csc x cot xdxd( tan x ) sec2 xdxd( cot x ) - csc2 xdxInverse Trig Functionsd1sin -1 x ) (dx1 - x2d( sec-1 x ) 12dxx x -1d1cos -1 x ) (dx1- x2d( csc-1 x ) - 12dxx x -1d1tan -1 x ) (dx1 x2d( cot -1 x ) - 1 1x2dxd n( x ) nxn-1dxd( cxn ) ncxn -1dxBasic Properties/Formulas/RulesÚ cf ( x ) dx c Ú f ( x ) dx , c is a constant.Ú f ( x ) g ( x ) dx Ú f ( x ) dx Ú g ( x ) dxÚ a f ( x ) dx F ( x ) a F (b ) - F ( a ) where F ( x ) Ú f ( x ) dxbbbbbbbÚ a cf ( x ) dx c Ú a f ( x ) dx , c is a constant. Ú a f ( x ) g ( x ) dx Ú a f ( x ) dx Úa g ( x ) dxabÚ a f ( x ) dx 0baÚ a f ( x ) dx -Úb f ( x ) dxcbbÚ a f ( x ) dx Ú a f ( x ) dx Úc f ( x ) dxIf f ( x ) 0 on a x b thenÚ a c dx c ( b - a )bÚ a f ( x ) dx 0If f ( x ) g ( x ) on a x b thenbbÚ a f ( x ) dx Ú a g ( x ) dxCommon IntegralsPolynomials1Ú dx x cÚ k dx k x cÚ x dx n 1 xÛ 1 dx ln x cÙıxÚxÚx-1ndx ln x cÛ 1 dx 1 ln ax b cÙı ax baÚxpq-ndx pq c, n -11x - n 1 c , n 1-n 1pdx n 11 q 1qx c x 1p qTrig FunctionsÚ cos u du sin u cp qq cÚ sin u du - cos u cÚ sec u du tan u cÚ sec u tan u du sec u c Ú csc u cot udu - csc u c Ú csc u du - cot u cÚ tan u du ln sec u cÚ cot u du ln sin u c1Ú sec u du ln sec u tan u cÚ sec u du 2 ( sec u tan u ln sec u tan u ) c223Exponential/Logarithm Functionsd xd x( a ) a x ln ( a )(e ) exdxdxdd( ln ( x ) ) 1x , x 0( ln x ) 1x , x 0dxdxHyperbolic Trig Functionsd( sinh x ) cosh xdxd( sech x ) - sech x tanh xdxÚ csc u du ln csc u - cot u cd( log a ( x ) ) x ln1 a , x 0dxVisit http://tutorial.math.lamar.edu for a complete set of Calculus I & II notes. 2005 Paul3u du 1( - csc u cot u ln csc u - cot u ) c2Exponential/Logarithm FunctionsuuÚ e du e cdd( cosh x ) sinh x( tanh x ) sech 2 xdxdxdd( csch x ) - csch x coth x( coth x ) - csch 2 xdxdxÚ cscuÚ a du au cln ae au( a sin ( bu ) - b cos ( bu ) ) ca b2e auauecosbudu ()( a cos ( bu ) b sin ( bu ) ) cÚa 2 b2ÚeDawkinsausin ( bu ) du 2Visit http://tutorial.math.lamar.edu for a complete set of Calculus I & II notes.Ú ln u du u ln ( u ) - u cÚ ue du ( u - 1) euu cÛ 1 du ln ln u cÙı u ln u 2005 PaulDawkins

Common Derivatives and IntegralsInverse Trig Functions1ÛÊu ˆdu sin -1 Á cÙ22Ëa ı a -uÚ sin1Û 1Êuˆdu tan -1 Á cÙ 22aı a uËa 11ÛÊuˆdu sec-1 Á cÙaËa ı u u2 - a2Hyperbolic Trig FunctionsÚ sinh u du cosh u cÚ sech tanh u du - sech u cÚ tanh u du ln ( cosh u ) c-1Ú tanu du u sin -1 u 1 - u 2 c1u du u tan -1 u - ln (1 u 2 ) c2-1Ú cos-1u du u cos - 1 u - 1 - u 2 cÚ cosh u du sinh u cÚ sechÚ csch coth u du - csch u c Ú cschÚ sech u du tan sinh u cMiscellaneousÛ 1 du 1 ln u a cÙ 2ı a - u22a u - aÚ2au - u 2 du Úu du tanh u c2u du - coth u cÛ 1 du 1 ln u - a cÙ 2ı u - a22a u aÚÚ2-1ua2a u du a 2 u 2 ln u a 2 u 2 c222uau 2 - a 2 du u 2 - a 2 - ln u u 2 - a 2 c22u 2a2Êuˆ222a - u du a - u sin -1 Á c22Ëa 2Common Derivatives and IntegralsFactor in Q ( x )2u SubstitutionÚ a f ( g ( x ) ) g ( x ) dx then the substitution u g ( x ) will convert this into thebg ( b)integral, Ú f ( g ( x ) ) g ( x ) dx Úf ( u ) du .ag ( a)bIntegration by PartsThe standard formulas for integration by parts are,Ú udv uv - Ú vdubbbÚa udv uv a - Úa vduChoose u and dv and then compute du by differentiating u and compute v by using thefact that v Ú dv .Visit http://tutorial.math.lamar.edu for a complete set of Calculus I & II notes.Term in P.F.D Factor in Q ( x )ax bAax bax 2 bx cAx Bax 2 bx c( ax b )( ax2Term in P.F.DA1A2Ak L kax b ( ax b ) 2( ax b )k bx c )Ak x BkA1 x B1 L k2ax bx c( ax2 bx c )kProducts and (some) Quotients of Trig FunctionsnmÚ sin x cos x dxu-aa2Ê a -u ˆ2 au - u 2 cos-1 Á c22Ë a Standard Integration TechniquesNote that all but the first one of these tend to be taught in a Calculus II class.GivenTrig SubstitutionsIf the integral contains the following root use the given substitution and formula.aa 2 - b2 x2fix sin qandcos2 q 1 - sin 2 qbab2x2 - a 2fix sec qandtan 2 q sec 2 q -1baa2 b2 x2fix tan qandsec 2 q 1 tan 2 qbPartial FractionsÛ P ( x)If integrating Ùdx where the degree (largest exponent) of P ( x ) is smaller than theı Q (x)degree of Q ( x ) then factor the denominator as completely as possible and find the partialfraction decomposition of the rational expression. Integrate the partial fractiondecomposition (P.F.D.). For each factor in the denominator we get term(s) in thedecomposition according to the following table. 2005 PaulDawkins1. If n is odd. Strip one sine out and convert the remaining sines to cosines usingsin 2 x 1 - cos2 x , then use the substitution u cos x2. If m is odd. Strip one cosine out and convert the remaining cosines to sinesusing cos 2 x 1 - sin 2 x , then use the substitution u sin x3. If n and m are both odd. Use either 1. or 2.4. If n and m are both even. Use double angle formula for sine and/or half angleformulas to reduce the integral into a form that can be integrated.ntanxsec m x dxÚ1. If n is odd. Strip one tangent and one secant out and convert the remainingtangents to secants using tan 2 x sec 2 x - 1 , then use the substitution u sec x2. If m is even. Strip two secants out and convert the remaining secants to tangentsusing sec 2 x 1 tan 2 x , then use the substitution u tan x3. If n is odd and m is even. Use either 1. or 2.4. If n is even and m is odd. Each integral will be dealt with differently.Convert Example : cos 6 x ( cos 2 x ) (1 - sin 2 x )33Visit http://tutorial.math.lamar.edu for a complete set of Calculus I & II notes. 2005 PaulDawkins

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Di erential Equations Study Guide1Second Order Linear EquationsGeneral Form of the EquationFirst Order Equationsdy f (x, y)dxInitial Value Problem: y 0 f (x, y), y(x0 ) y0(1)General Form of ODE:(2)(22)General Form: a(t)y 00 b(t)y 0 c(t)y g(t)(23)00Homogeneous: a(t)y b(t)y c(t) 0(24)Standard Form: y 00 p(t)y 0 q(t)y f (t)(3)(4)(5)(6)General Form: y 0 p(x)y f (x)Rp(x)dxIntegrating Factor: µ(x) ed )(µ(x)y) µ(x)f (x)dx Z 1General Solution: y µ(x)f (x)dx Cµ(x)Homeogeneous Equations(7)(8)General Form: y f (y/x)Substitution: y zx ) y 0 z xz 0(9)@2. Set N (x, y)@y3. Simplify and solve for h(y).4. Substitute the result for h(y) in the expression for1 and then set 0. This is the solution.from stepAlternatively:R1. Let N (x, y)dy g(x)(26)dzdx f (z) zxBernoulli Equations(11)(12)General Form: y 0 p(x)y q(x)y nSubstitution: z y 1n(18)P (x, y) My2. {y1 , y2 } are a fundamental set of solutions.Cauchy-Euler EquationNxN ) µ(y) eRP (x)dxz 0 (1n)p(x)z (1thenn)q(x)(19)(28)µ(x)M (x, y)dx µ(x)N (x, y)dy 0Case 2: If Q(x, y) depends only on y, where(14)(20)(15)(16)(17)Q(x, y) NxMyM ) µ(y) e(29)RQ(y)dyThen(21)(30)(31)(32)µ(y)M (x, y)dx µ(y)N (x, y)dy 0is exact.1p(t)dt(40)2013 http://integral-table.com. This work is licensed under the Creative Commons Attribution – Noncommercial – No Derivative Works 3.0 United StatesLicense. To view a copy of this license, visi

Trig Cheat Sheet Definition of the Trig Functions t Right triangle definition For this definition we assume that 0 2 p q or 0 q 90 . 11 opposite sin hypotenuse q hypotenuse csc opposite q adjacent cos hypotenuse q hypotenuse sec adjacent q opposite tan adjacent q adjacent cot opposite q P Unit circle definition For this definition q is any angle. sin 1 y q y 1 csc y q (cos 1 x q .