Maths basic useful formulas, Study notes of Family and Consumer Science

Download Maths basic useful formulas and more Study notes Family and Consumer Science in PDF only on Docsity! Applied Mathematics - I! [ 1. Trigonometic F 1. Definitions of Trigonometric Ratios (F-1) 2. Fundamental Trigonometic Identities (A) sin (- 6) =- sin, sin ( - 0)- cos 8, 2 oin( E + 0) = cos 0, 2 sin (x — 6) = sin 6, sin (1 + 6) = — sin 8, sino] = —cos 6, 2 an( + e} =—-cos@, (B) Quadrant sind j + Ul + til - IV - cos (— 6) = cos 6 cos (5 - 0) = sin®, 2 nt cos|—+6|=—sin6, ( ) in 8, cos (1-6) =—cos 8, cos (n+ 6) =—cos 8, cos - 0) = —sin6, cos (= + 0) = sing, cos6 = tand + + - + + = List of Formul ormulae Scanned by CamScanner mee oS (c) (D) (©) (F) S270, 180° = 7° a2 sin 0 + c05? 0 = 1, sec? 6 = 1 + tan9, cosec® 6 = 1 + cot” 6. sin (a + B) = sin a cos B + cos o sin B cos (a + B) = cos a cos B F sina sinB sin 20 = 2 sin @ cos 0 cos 20 = cos 0~sin? 9 = 2cos’0-1 = 1-2sin"0 sin 30 =3sin0—4 sin°0 cos 30= 4 cos’ 0 - 3cos 0 2 tand t= tan? @ si —_— 0s 20 = in 26 1+ tan? @ cos 1+ tan? 0 2tan@ gq = 21and= tan tan 26 =~ 2@ tan 90 = "1" 3 tan? 0 c-D sinC + sinD = 2sin =? cos —F— +D.C-D sin - sin D = 2608 5== sin 2 | ! c+D 6-0 cos C + cos D = 2c0s—5— 0S 2 c-D —_—_— D. cos C- cos D = -28in = sin 2 ——— —_—— _ _ Scanned by CamScanner atics «I (F5) List of Formulae Appiled Mathem each element of a ow or column is multiplied by a constant then id) If eac! . . the ie rminant ig multiplied by that constant ¢ equimukipies of (e) The value of a determinant is unchanged if eq Pp @ row or 8 column are added to the corresponding elements of any other row or column, a Cramer’s Rule fay x byt 2 ap X+ Dy V+ G2 2= Ob ag X+ Dg y+ CgZ= Oy, Dn ype za x y= D’ Zz D then x =D a, by Cy where, D=|a bo ¢2 ag bg 63 and D,, Dy Dz are obtained by replacing the coefficients of x, y, Z respectively by a), d, dy. 3. Differentiation Formulae _ (sine f 1y (A) lim | ——|=1, lm }1+—| =e, | 9-0\ 6 nse on x lim (1+ y)¥ = 8, lim 2 = = . Pav y) x70 Xx Gea (B) 1. Ify=x?, YS nyt ax 2. If y=sin x, Y _ cog = x 3. If y= cos x, Y _ _ sin ck x 4. ity=tanx, ® - sec? cx sec” x 5. If y=cosec x, ® ~~ cosec xcot x ax 6. If y= sec x, & « sec xtanx 7. ify =cotx, ® _ _ cosec? ce 7 COseC™ x 8. Ify= Y 6 y aw? — Scanned by CamScanner Applied Mathematics - Il 9. 10, 11. 12. 13. 14, 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. Ify= a’, If y= log, x, If y= log, x, ity=sin” x, lf y=cos™ x, Ify=tan” x, If y=sec™ x, lf y=cosec” x, If y= cot’ x, Ify=sin hx, * If y= cos hx, If y= tan hx, If y= cosec hx, If y= sec hx, If y= cot hx, y= sin x, If y= cos Ht x, If y= tan tr x, If y=sec H x, FR a £ List of Formulae ui s 3 2 It xf ~ Hi = = a 1 fig — & & ny a 1 = I * nN 4 I ? >G] = Ao 1 HW * xe 1 2 gle ge gle sie gle g[e gle gle Bd x | = g| i im : *p & = coshx ad. —-=sinh ax IN AX gy 2 —=sech ox che x ay Ye -cosec h xcot x dx d Y sechxtanhx dx yy =~cosec h? x dx Ys A ax 442 gy _ 1 ow yx? -1 yt dx 1—x* [on a xvi- 2 Scanned by CamScanner Applied Mathematics - Il 28. If y=cosec Hx, 29. If y= cot A"! x, (C) 1. Ifysusy, 2. Ify= uv, 5. If x= f(t), y= (t), 6. 8. (F-7) 0 ~ [xIvt + x? _ a T eo a 2 Sle H g|2 wv ax dv u— ox " *, | + gl2& I Wt | glesie gigeie gig gis = x*(1+logx) dy _ dy/dt dx dx/dt List of Formulas 4. Integration Formulae fi-U-ax=1. fr x |[futae]- SE oe fer Ax + F(a lax = & f(x) n+ x x" dx = ifn*-1 n+1 ax = heox 3. J sin x dx = -cos x cos x dx = sin x 5. Jsec x dx = tan x 2 cosec* x dx = — cot x 7. J sec x tan x dx = sec x cosec x cot x dx = — cosec x 9. J tan x ax = logsec x 10. J cot x ax = -logcosec x = logsin x 11. 12. 13. secxox=1og{tan( +2) = log(sec x + tan x) cosec x dx = loa tan ) = log(cosec x — cot x) e* dx = @* 14. fa*ax = _ oe = sin” io0a 15. j Te & [>< Scanned by CamScanner 2. Exponential Limits . We use the series e* +lrer st te e wh (i) lim 1 x0 x r -_ iii Gea x0 = log, a dx (iii) im “— ly where (A #0). — x 3. Logarithmic Limits We use the series log(] + x)= x - a x FS vee 00 3 where —-1< x< 1] and expansion is true only, if base is e. (i) lim 2080.4) _ | "£40 xX (ii) lim log, x=1 i—¢ au) lim log, (1 - x) =-j] rO0 x (iv) lim log, (1+ 2) = log, e,a>0,#1 x0 r (v) If im f(x) exists and positive, then i-a 8 Lim @(x) log f(x) lim [f(x)}* = e7=* 14a Scanned by CamScanner 4. Based on the Form 1° To evaluate the exponential farm 17, we use following results. If hm f(x) = lim g(x) = 0, ra lim [2 then, lim {14 f(x)p¥#lt) — pt oeetx) ra or when lim f(x)=1 and lim g(x)=&, ta x40 i lam | f(#) - Then, lim { f(axyp 8? = lim {14 fix)- 1} Azeem! x)-1) gtx ra ra 1 (i) Lim (1+ x)" =e x0 l x (ui) lim l+ ) =e r= x \ (iii) lim (1+ Ax} =e zt x (iv) lim (2 +4) <¢ ie x 0, O<a<l 1 a=l1 (v) lim a’* = r# oo a>l does not exist, a<0 Scanned by CamScanner  Important formulae: A+x) (1-x)" (14x)? il tl ll 1=x +x? X84 1+ x4+x? 4x3 40... 1-2x + 3x? — 4x34 14+ 2x+ 3x? + 4x? 4... Scanned by CamScanner