Trig Cheat Sheet. Formulas and Identities Tangent and Cotangent Identities

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1 Trig Chet Sheet Definition of the Trig Funtions Right tringle efinition For this efinition e ssume tht Unit irle efinition For this efinition is n ngle. 0 < < or 0 < < 90. oosite oosite sin hotenuse jent os hotenuse oosite tn jent hotenuse jent hotenuse s oosite hotenuse se jent jent ot oosite (, sin s os se tn ot Fts n Proerties Domin The omin is ll the vlues of tht Perio n e lugge into the funtion. The erio of funtion is the numer, T, suh tht f ( + T f (. So, if sin, n e n ngle is fie numer n is n ngle e os, n e n ngle hve the folloing erios. Ê tn, π Án+, n 0, ±, ±, sin ( Æ T s, π n, n 0, ±, ±, Ê se, π Án+, n 0, ±, ±, os( Æ T ot, π n, n 0, ±, ±, tn ( Æ T Rnge s( Æ T The rnge is ll ossile vlues to get out of the funtion. - sin s n s - se( Æ T - os se nse - - < tn < - < ot < ot ( Æ T Formuls n Ientities Tngent n Cotngent Ientities Hlf Angle Formuls sin os tn ot sin ( - os( os sin Reirol Ientities os ( + os( s sin sin s - os( tn se os + os( os se Sum n Differene Formuls ot tn sin ( ± sin os ± ossin tn ot Pthgoren Ientities os( ± os os m sinsin sin + os tn ± tn tn ( ± tn + se m tn tn Prout to Sum Formuls + ot s Even/O Formuls sin sin Èos( os( Î sin (- -sin s( - - s os os Èos( os( os( - os se( - se Î tn (- - tn ot (- -ot sin os Èsin ( sin ( Î Perioi Formuls If n is n integer. os sin Èsin ( sin ( Î sin ( + n sin s( + n s Sum to Prout Formuls os( + n os se( + n se Ê + Ê - sin + sin sin Á osá tn ( + n tn ot ( + n ot Doule Angle Formuls Ê + Ê - sin - sin osá sin Á sin ( sinos Ê + Ê - os( os - sin os + os osá osá os - Ê + Ê - os - os -sin sin - sin Á Á tn Cofuntion Formuls tn( - tn Degrees to Rins Formuls If is n ngle in egrees n t is n ngle in rins then t 80 fi t n t Ê Ê sin Á - os osá - sin Ê Ê sá - se seá - s Ê Ê tn Á - ot ot Á - tn

2 Ê Á- Ê Á-, (-,0, 5 6 Ê Á-, Unit Cirle ( 0, Ê Á, 45 4 Ê Á , Ê Á 0, (,0 Inverse Trig Funtions Definition Inverse Proerties sin is euivlent to sin os os ( os os - os is euivlent to os - tn is euivlent to tn Domin n Rnge Funtion Domin Rnge - sin os tn - < < - < < ( ( ( ( -( -( ( ( -( -( ( sin sin sin sin tn tn tn tn Alternte Nottion - sin rsin os tn - - ros rtn L of Sines, Cosines n Tngents Ê Á-,- or n orere ir on the unit irle (, mle 7 6 Ê Á-, Ê Á-, : os n sin Ê 5 Ê5 osá sin Á Ê 5 Á, Ê Á,- Ê Á,- ( 0, - L of Sines sin sin sing L of Cosines + - os os osg Molleie s Formul + os ( - sin g g L of Tngents - tn - + tn + ( ( ( - g ( g ( -g ( g - tn + tn + - tn + tn +

3 Common Derivtives n Integrls Common Derivtives n Integrls Derivtives Bsi Proerties/Formuls/Rules ( f ( f (, is n onstnt. ( f ( ± g ( f ( ± g ( n n ( n -, n is n numer. ( 0, is n onstnt. ( f g Ê f f g - fg f g + fg (Prout Rule Á (Quotient Rule g g ( f ( g ( f ( g ( g ( (Chin Rule g( g ( g ( ( e g ( e ( ln g( g Common Derivtives Polnomils ( 0 ( ( Trig Funtions ( sin os ( se se tn n ( n n n- n n- ( ( ( os - sin ( tn se ( s - s ot ( ot - s Inverse Trig Funtions - ( sin - - ( se - - ( os ( s - - Eonentil/Logrithm Funtions ( ln ( ( e e ( ln (, > 0 ( ln, 0 Heroli Trig Funtions ( sinh osh ( seh seh tnh - ( tn - ( ot ( ( log, > 0 ln ( osh sinh ( tnh seh - ( sh - sh oth ( oth - sh Bsi Proerties/Formuls/Rules f f Integrls ( (, is onstnt. ( ± ( ( ± ( ( ( ( - ( here F ( f ( f F F F f g f g f ( f (, is onstnt. ( ± ( ( ± ( f g f g ( 0 ( - ( f f f f ( f ( + f ( ( - f If f ( 0 on then ( 0 If f ( g( on then ( ( Common Integrls Polnomils + k k + Û Ù ln + ı Û Ù ln + + ı + - ln + f g n n+ +, n - n + -n - n+ +, n - n Trig Funtions os u u sin u + sin u u - osu+ se u u tn u+ se u tn u u se u + s u ot uu - su+ s u u - ot u+ tn u u ln se u + ot u u ln sin u + se u u ln se u + tn u + se u u ( se u tn u + ln se u + tn u + s u u ln s u - ot u + s u u (- s uot u + ln su - ot u + Eonentil/Logrithm Funtions u u u e u e u + u + ln u e + u u e e os ( u u ( os( u + sin ( u + + ln ln ( u u u u - u+ u u e sin( u u ( sin( u - os( u + ( e e u u u u- + Û Ù ln ln uln u u u + ı Visit htt://tutoril.mth.lmr.eu for omlete set of Clulus I & II notes. 005 Pul Dkins Visit htt://tutoril.mth.lmr.eu for omlete set of Clulus I & II notes. 005 Pul Dkins

4 Common Derivtives n Integrls Common Derivtives n Integrls Inverse Trig Funtions Û - Êu Ù u sin Á + ı - u Û u Ù u tn - Ê Á + ı + u Û se - Êu Ù u Á + ı u u sin u u usin u + - u tn u u u tn u - ln( + u os u u u os u - - u + Heroli Trig Funtions sinh u u osh u + osh u u sinh u+ seh tnh u u - seh u + sh oth u u - sh u+ - tnh u u ln osh u + seh u u tn sinh u + ( Misellneous Û u+ Ù u ln + ı -u u- Û Ù ı u + u u + u + ln u + + u + u u - u u - - ln u + u u u u - u + sin u - Ê Á + u - -Ê-u u - u u u - u + os Á + seh sh u - u+ u u tnh u+ u- u ln + u u - oth u+ Stnr Integrtion Tehniues Note tht ll ut the first one of these ten to e tught in Clulus II lss. u Sustitution Given ( ( ( then the sustitution u g( f g g g integrl, f ( g ( g ( f ( u u. ( g ( Integrtion Prts The stnr formuls for integrtion rts re, - - ill onvert this into the uv uv vu uv uv vu Choose u n v n then omute u ifferentiting u n omute v using the ft tht v v. Trig Sustitutions If the integrl ontins the folloing root use the given sustitution n formul. - fi sin n os - sin - fi se n tn se - + fi tn n se + tn Prtil Frtions Û P( If integrting Ù ı Q( here the egree (lrgest eonent of P( is smller thn the egree of Q( then ftor the enomintor s omletel s ossile n fin the rtil frtion eomosition of the rtionl eression. Integrte the rtil frtion eomosition (P.F.D.. For eh ftor in the enomintor e get term(s in the eomosition oring to the folloing tle. Ftor in Q( Term in P.F.D Ftor in Q( + A + A+ B + + ( + k ( Prouts n (some Quotients of Trig Funtions n m sin os k Term in P.F.D A A Ak + + L ( ( A+ B Ak+ Bk + L (. If n is o. Stri one sine out n onvert the remining sines to osines using sin - os, then use the sustitution u os. If m is o. Stri one osine out n onvert the remining osines to sines using os - sin, then use the sustitution u sin. If n n m re oth o. Use either. or. 4. If n n m re oth even. Use oule ngle formul for sine n/or hlf ngle formuls to reue the integrl into form tht n e integrte. n m tn se. If n is o. Stri one tngent n one sent out n onvert the remining tngents to sents using tn se -, then use the sustitution u se. If m is even. Stri to sents out n onvert the remining sents to tngents using se + tn, then use the sustitution u tn. If n is o n m is even. Use either. or. 4. If n is even n m is o. Eh integrl ill e elt ith ifferentl. 6 Convert Emle : os ( os ( - sin k k Visit htt://tutoril.mth.lmr.eu for omlete set of Clulus I & II notes. 005 Pul Dkins Visit htt://tutoril.mth.lmr.eu for omlete set of Clulus I & II notes. 005 Pul Dkins

5 ] U X ] X ρ,,] Xρ ] G] ρ Gρ Uos θ ] θ U X U X Xθ ] θ Gθ U GU Usin θ UGθ ρ ρ ρ Usin θ Usin θ Ω os ' Ω + Ω ' ' rtn(/ z z z z r µ os ' r µ ' r + + z µ rtn( + /z z r os µ ' rtn(/ u Ω os'u + 'u u ' 'u +os'u u z u z u r µ os 'u + µ 'u +osµu z u µ µ os 'u + µ 'u µu z u ' 'u +os'u r Ωu Ω + zu z r ru r r l Ω u Ω + l ' u ' + l z u z Ωu Ω + Ω'u ' + zu z r l r u r + l µ u µ + l ' u ' ru r + rµu µ + r µ'u ' S Ω l ' l z Ω'z ; S ' l Ω l z Ωz ; S z l Ω l ' ΩΩ' S r l µ ' r µµ' ; S µ l r l ' r µr' ; S ' l r l µ rrµ ø l Ω l ' l z ΩΩ'z ø l r l µ ' r µrµ'

6 A ' (r, µ, 'u ' f f(r, µ, ' A(r, µ, ' A r (r, µ, 'u r + A µ (r, µ, 'u µ + A z (,, zu z f f(,, z A(,, z A (,, zu + A (,, zu u z @ z @A @A! @A! u r (rf r @r u r u µ r u ' r A r µa µ ' r µa' r µ u r r + r u r µ f r u µ A B C A z (Ω, ',zu z f f(ω, ',z A(Ω, ',za Ω (Ω, ',zu Ω +A ' (Ω, ',zu ' u Ω u u z r Ω ' z z r u @A! z u + Ω u r A (B C B (C A (A B C A (B C B(A C C(A B f f(r g g(r A A(r B B(r r(f + g rf + rg r(fg f(rg+g(rf r (A + B r A + r B r (fa f(r A+A (rf r (A B B (r A A (r B r (A + B r A + r B r (fa f(r A A (rf r (r A 0 r (rf 0 r (r A r(r A r A

7 Liner Eutions ( (4 (5 (6 Generl Form: 0 + ( f( Integrting Ftor: µ( e R ( Di erentil Eutions Stu Guie First Orer Eutions ( Generl Form of ODE: f(, ( Initil Vlue Prolem: 0 f(,, (0 0 (µ( µ(f( Generl Solution: Z µ(f( + C µ( Homeogeneous Eutions (7 (8 (9 Generl Form: 0 f(/ Sustitution: z 0 z + z 0 The result is ls serle in z: (0 Bernoulli Eutions ( ( z f(z z Generl Form: 0 + ( ( n Sustitution: z n The result is ls liner in z: ( z 0 +( n(z ( n( Et Eutions (4 (5 (6 (7 Generl Form: M(, + N(, 0 Tet Solution: C Metho for Solving Et Eutions:. Let R M(, + h(. Simlif n solve for h(. 4. Sustitute the result for h( in the eression for from ste n then set 0. This is the solution. Alterntivel:. Let R N(, + g(. Simlif n solve for g(. 4. Sustitute the result for g( in the eression for from ste n then set 0. This is the solution. Integrting Ftors Cse : If P (, eens onl on, here M N (8 P (, µ( e R P ( N then (9 µ(m(, + µ(n(, 0 is et. Cse : If Q(, eens onl on, here N M (0 Q(, M Then µ( e R Q( ( µ(m(, + µ(n(, 0 is et. 0 htt://integrl-tle.om. This ork is liense uner the Cretive Commons Attriution Nonommeril No Derivtive Works.0 Unite Sttes Liense. To vie o of this liense, visit: htt://retiveommons.org/lienses/-n-n/.0/us/. This oument is rovie in the hoe tht it ill e useful ut ithout n rrnt, ithout even the imlie rrnt of merhntilit or fitness for rtiulr urose, is rovie on n s is sis, n the uthor hs no oligtions to rovie orretions or moifitions. The uthor mkes no lims s to the ur of this oument, n it m ontin errors. In no event shll the uthor e lile to n rt for iret, iniret, seil, inientl, or onseuentil mges, inluing lost rofits, unstisftor lss erformne, oor gres, onfusion, misunerstning, emotionl isturne or other generl mlise rising out of the use of this oument, even if the uthor hs een vise of the ossiilit of suh mge. This oument is rovie free of hrge n ou shoul not hve i to otin n unloke PDF file. Revise: Jul, 0. Generl Form of the Eution ( ( (4 Generl Form: (t 00 + (t 0 + (t g(t Homogeneous: (t 00 + (t 0 + (t 0 Stnr Form: 00 + (t 0 + (t f(t The generl solution of ( or (4 is (5 C (t+c (t+ (t here (t n (t re linerl ineenent solutions of (. Liner Ineenene n The Wronskin To funtions f( n g( re linerl eenent if there eist numers n, not oth zero, suh tht f(+g( 0 for ll. If no suh numers eist then the re linerl ineenent. If n re to solutions of ( then (6 (7 Wronskin: W (t (t(t 0 (t 0 (t Ael s Formul: W (t Ce R (tt n the folloing re ll euivlent:. {, } re linerl ineenent.. {, } re funmentl set of solutions.. W (, (t t some oint t W (, (t 6 0 for ll t. Initil Vlue Prolem (8 8 < 00 + (t 0 + (t 0 (t 0 0 : 0 (t 0 Liner Eution: Constnt Coe (9 (0 ( ( Seon Orer Liner Eutions ients Homogeneous: Non-homogeneous: g(t Chrteristi Eution: r + r + 0 The solution of (9 is given : ( (4 (5 Qurti Roots: r ± 4 Rel Roots(r 6 r : H C e rt + C e rt Reete(r r : H (C + C te rt Comle(r ± i : H e t (C os t + C sin t The solution of (0 is P + h H here h is given ( through (5 n P is foun unetermine oe ients or reution of orer. Heuristis for Unetermine Coe ients (Tril n Error If f(t then guess tht P Pn(t t s (A0 + At + + Ant n Pn(te t t s (A0 + At + + Ant n e t Pn(te t sin t t s e t (A0 + At + + Ant n ost or Pn(te t os t +(A0 + At + + Ant n sint] Metho of Reution of Orer When solving (, given,then n e foun solving (6 0 0 Ce R (tt The solution is given (7 Z e R ( ( Metho of Vrition of Prmeters If (t n (t re funmentl set of solutions to ( then rtiulr solution to (4 is Z Z (tf(t (tf(t (8 P (t (t t + (t t W (t W (t Cuh-Euler Eution (9 (40 ODE: Auillir Eution: r(r + r + 0 The solutions of (9 een on the roots of (40: (4 (4 (4 Rel Roots: C r + C r Reete Root: C r + C r ln Series Solutions Comle: C os( ln +C sin( ln ] (44 ( ( 0( 0 + ( 0 If 0 is regulr oint of (44 then X (45 (t ( 0 n k( k0 At Regulr Singulr Point 0: (46 (47 k k Iniil Eution: r +((0 r + (0 0 X First Solution: ( 0 r k( k k k0 Where r is the lrger rel root if oth roots of (46 re rel or either root if the solutions re omle.

Calculus Cheat Sheet. Integrals Definitions. where F( x ) is an anti-derivative of f ( x ). Fundamental Theorem of Calculus. dx = f x dx g x dx

Calculus Cheat Sheet. Integrals Definitions. where F( x ) is an anti-derivative of f ( x ). Fundamental Theorem of Calculus. dx = f x dx g x dx Clulus Chet Sheet Integrls Definitions Definite Integrl: Suppose f ( ) is ontinuous Anti-Derivtive : An nti-derivtive of f ( ) on [, ]. Divide [, ] into n suintervls of is funtion, F( ), suh tht F = f.