SOME IMPORTANT MATHEMATICAL FORMULAE

Transcription

1 SOME IMPORTANT MATHEMATICAL FORMULAE Circle : Are = π r ; Circuferece = π r Squre : Are = ; Perieter = 4 Rectgle: Are = y ; Perieter = (+y) Trigle : Are = (bse)(height) ; Perieter = +b+c Are of equilterl trigle = 4 Sphere : Surfce Are = 4 π r ; Volue = 4 π r Cube : Surfce Are = 6 ; Volue = Coe : Curved Surfce Are = π rl ; Volue = π r h Totl surfce re = π r l + π r Cuboid : Totl surfce re = (b + bh + lh); Volue = lbh Cylider : Curved surfce re = π rh; Volue = π r h Totl surfce re (ope) = π rh; Totl surfce re (closed) = π rh+ π r SOME BASIC ALGEBRAIC FORMULAE: ( + b) = + b+ b ( - b) = - b+ b ( + b) = + b + b( + b) 4 ( - b) = - b - b( - b) 5( + b + c) = + b + c +b+bc +c 6( + b + c) = + b + c + b+ c + b c +b +c +c +6bc 7 - b = ( + b)( b ) 8 b = ( b) ( + b + b ) 9 + b = ( + b) ( - b + b ) ( + b) + ( - b) = 4b ( + b) - ( - b) = ( + b ) If + b +c =, the + b + c = bc INDICES AND SURDS = + = ( ) = (b) = b 4 5 = b b 6 =, 7 = 8 y = = y 9 = b = b ± b = ± y, where + y = d y = b M Sc, MIE, M Phil

2 LOGARITHMS = log = ( > d ) log = log + log log = log log log = log 4 log b = log log b 5 log = 6 log = 7 log b = log b 8 log = 9 log ( +) log +log e log = log = PROGRESSIONS ARITHMETIC PROGRESSION, + d, +d, re i AP th ter, T = + (-)d Su to ters, S = [ + ( )d ] If, b, c re i AP, the b = + c GEOMETRIC PROGRESSION, r, r, re i GP ( r ) (r ) Su to ters, S = if r < d S = if r > r r Su to ifiite ters of GP, S = r If, b, c re i AP, the b = c HARMONIC PROGRESSION Reciprocls of the ters of AP re i HP,,, re i HP + d + d If, b, c re i HP, the b = c + c MATHEMATICAL INDUCTION ( + ) = = ( + )( + ) = = 6 M Sc, MIE, M Phil

3 ( + ) = = 4 PERMUTATIONS AND COMBINATION! r! P r = ( )! r! r! C = r ( )!= C r = C -r C r + C r- = ( + ) C r ( + )! ( + )C r =!! BINOMIAL THEOREM ( +) = + C - + C - + C C th ter, T r+ = C r -r r PARTIAL FRACTIONS f () is proper frctio if the deg (g()) > deg (f()) g() f () is iproper frctio if the deg (g()) deg (f()) g() Lier o- repeted fctors f () A B = + ( + b)(c + d) + b (c + d) Lier repeted fctors f () A B C = + + ( + b)(c + d) + b (c + d) (c + d) No-lier(qudrtic which c ot be fctorized) f () A + B C + D = + ( + b)(c + d) + b (c + d) ANALYTICAL GEOMETRY Distce betwee the two poits (, y ) d (, y ) i the ple is ( ) + (y y ) OR Sectio forul + y + y, + + y y, (for iterl divisio), (for eterl divisio) ( ) + (y y ) M Sc, MIE, M Phil

4 4 Mid poit forul + y + y, 4 Cetriod forul + + y + y + y, 5 Are of trigle whe their vertices re give, (y y ) = [ (y y ) + (y y ) + (y y ) ] STRAIGHT LINE Slope (or Grdiet) of lie = tget of iclitio = tθ Slope of X- is = Slope of lie prllel to X-is = Slope of Y- is = Slope of lie prllel to Y-is = y y Slope of lie joiig (, ) d (y, y ) = If two lies re prllel, the their slopes re equl ( = ) If two lies re perpediculr, the their product of slopes is - ( = -) EQUATIONS OF STRAIGHT LINE y = + c (slope-itercept for) y - y = (- ) (poit-slope for) y y y y = ( ) (two poit for) y + = (itercept for) b cosα +y siα = P (orl for) Equtio of stright lie i the geerl for is + b + c = Slope of + b + c = is b Agle betwee two stright lies is give by, tθ = + Legth of the perpediculr fro poit (, ) d the stright lie + b + c + by + c = is + b M Sc, MIE, M Phil

5 5 Equtio of stright lie pssig through itersectio of two lies + b + c = d + b + c = is + b + c + K( + b + c ) =, where K is y costt Two lies eetig poit re clled itersectig lies More th two lies eetig poit re clled cocurret lies Equtio of bisector of gle betwee the lies + b y+ c = d + by + c + by + c + b y + c = is = ± + b + b PAIR OF STRAIGHT LINES A equtio +hy +by =, represets pir of lies pssig through origi geerlly clled s hoogeeous equtio of degree i d y d gle betwee these is give by tθ = h b + b +hy +by =, represets pir of coicidet lies, if h = b d the se represets pir of perpediculr lies, if + b = If d re the slopes of the lies +hy +by =,the + = h b d = b A equtio +hy +by +g +fy +c = is clled secod geerl secod order equtio represets pir of lies if it stisfies the the coditio bc + fgh f bg ch = The gle betwee the lies +hy +by +g +fy +c = is give by tθ = h b + b +hy +by +g +fy +c =, represets pir of prllel lies, if h = b d f = bg d the distce betwee the prllel lies is g c ( + b) +hy +by +g +fy +c =, represets pir of perpediculr lies,if + b = M Sc, MIE, M Phil

6 6 TRIGNOMETRY Are of sector of circle = r θ Arc legth, S = r θ siθ = opp dj opp dj hyp hyp,cosθ =,tθ =,cotθ =, secθ =, cosecθ = hyp hyp dj opp dj opp Siθ = cos ecθ or cosecθ = si θ, cosθ = secθ or secθ = cos θ, tθ = cot θ or cotθ = si θ cos θ, tθ =, cotθ = t θ cos θ si θ si θ + cos θ = ; si θ = - cos θ; cos θ = - si θ; sec θ - t θ = ; sec θ = + t θ; t θ = sec θ ; cosec θ - cot θ = ; cosec θ = + cot θ; cot θ = cosec θ STANDARD ANGLES π or or 6 45 or π 6 4 or π π π 9 or 5 or 75 or 5 π Si Cos T Cot Sec Cosec ALLIED ANGLES Trigooetric fuctios of gles which re i the d, rd d 4 th qudrts c be obtied s follows : If the trsfortio begis t 9 or 7, the trigooetric fuctios chges s si cos t cot sec cosec M Sc, MIE, M Phil

7 7 where s the trsfortio begis t 8 or 6, the se trigooetric fuctios will be retied, however the sigs (+ or -) of the fuctios decides ASTC rule COMPOUND ANGLES Si(A+B)=siAcosB+cosAsiB Si(A-B)= siacosb-cosasib Cos(A+B)=cosAcosB-siAsiB Cos(A-B)=cosAcosB+siAsiB t(a+b)= t A + t B t A t B t(a-b)= t A t B + t A t B π + t A t + A = 4 t A π t A t A = 4 + t A t A + t B + t C t A t B t C t(a+b+c)= (t A t B + t B t C + t C t A) si(a+b) si(a-b)= si A si B = cos B cos A cos(a+b) cos(a-b)= cos A si B MULTIPLE ANGLES t A si A= sia cosa si A= + t A cos A = cos A si A =-si A = cos A t A = + t A t A 4 t A= t A, 5 +cos A= cos A, 6 cos A = ( + cos A) 7 -cos A= si A, 8 si A = ( cos A), 9+si A= -si A= (cos A si A) = (si A cos A), cos A= t A t A si A= si A 4si A, t A= t A (si A cos A) 4cos A +, cos A, M Sc, MIE, M Phil

8 8 HALF ANGLE FORMULAE θ t θ θ ) si θ= si cos ) si θ= θ + t 4) θ cos θ = si 5) θ t 7) t θ = 8) θ t PRODUCT TO SUM θ θ = 6) cos cos sia cosb = si(a+b) + si(a-b) cosa sib = si(a+b) si(a-b) cosa cosb = cos(a+b) + cos(a-b) sia sib = cos(a+b) cos(a-b) θ + cos θ = cos 9) SUM TO PRODUCT C + D C D Si C + si D = si cos C D C D Si C si D = cos + si C D C D Cos C + cos D = cos + cos C D C D Cos C- cos D = si + si OR D C D C Cos C- cos D = si + si ) cos θ = cos t cosθ = t θ θ si θ θ + θ cos θ = si PROPERTIES AND SOLUTIONS OF TRIANGLE b c Sie Rule: = = = R, where R is the circu rdius of the si A si B si C trigle b + c Cosie Rule: = b + c -bc cosa or cosa =, bc M Sc, MIE, M Phil

9 9 + c b b = + c -c cosb or cosb =, c + b c c = + b -b cosc or cosc = b Projectio Rule: = b cosc +c cosb b = c cosa + cosc c = cosb +b cosa Tgets Rule: B C b c A t = cot b + c, C A c B t = cot c +, A B b C t = cot + b Hlf gle forul: A (s b)(s c) A s(s ) A (s b)(s c) si =, cos =, t = bc bc s(s ) B (s )(s c) B s(s b) si =, cos =, c c C (s )(s b) C s(s c) si =, cos =, b b Are of trigle ABC = s(s )(s b)(s c), Are of trigle ABC = bcsi A = csi B = bsi C LIMITS If f ( ) = f ( ), the f ( ) is clled Eve Fuctio If f ( ) = f ( ), the ( ) If P is the sllest ve periodic fuctio with period P f is clled Odd Fuctio B (s )(s c) t = s(s b) C (s )(s b) t = s(s c) + rel uber such tht if f ( + P ) = f ( ), the ( ) 4 Right Hd Liit (RHL) = li ( f ( ) ) = li ( f ( + h ) ) + h Left Hd Liit (LHL) = li ( f ( ) ) = li ( f ( h ) ) If RHL=LHL the li ( ( ) ) li ( ( ) ) f = RHL=LHL h f eists d f is clled M Sc, MIE, M Phil

10 5 Lt p =, if p > d Lt p = if p > si t Lt = Lt i rdis = Lt = Lt = si t 6 ( ) 7 si Lt si 8 Lt = π π t π = Lt = 8 9 si t li = = li li =, where is iteger or frctio e li = log, li = log e = li + = e, li + = e ( ) li kf ( ) = k li f ( ) 4 li f ( ) ± g ( ) = li f ( ) ± li g ( ) 5 li f ( ) g ( ) = li f ( ) li g ( ) ( ) ( ) ( ) li f li li ( ) f = provided g ( ) g li g 6 A fuctio ( ) f is sid to be cotiuous t the poit = if (i) li f ( ) eists (ii) f ( ) is defied (iii) li f ( ) = f ( ) 7 A fuctio f ( ) is sid to be discotiuous or ot cotiuous t (i) f ( ) is ot defied t (iii) li f ( ) li f ( ) f ( ) + = (ii) li f ( ) does ot eist t = 8 If two fuctios f ( ) d g ( ) re cotiuous the f ( ) g ( ) = if + is cotiuous M Sc, MIE, M Phil