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University Physics Volume 3

E | Mathematical Formulas

University Physics Volume 3E | Mathematical Formulas

Quadratic formula

If ax2+bx+c=0,ax2+bx+c=0, then x=−b±b2−4ac2ax=−b±b2−4ac2a

Triangle of base bb and height hh Area =12bh=12bh
Circle of radius rr Circumference =2Ï€r=2Ï€r Area =Ï€r2=Ï€r2
Sphere of radius rr Surface area =4Ï€r2=4Ï€r2 Volume =43Ï€r3=43Ï€r3
Cylinder of radius rr and height hh Area of curved surface =2Ï€rh=2Ï€rh Volume =Ï€r2h=Ï€r2h
Table E1 Geometry

Trigonometry

Trigonometric Identities

  1. sinθ=1/cscθsinθ=1/cscθ
  2. cosθ=1/secθcosθ=1/secθ
  3. tanθ=1/cotθtanθ=1/cotθ
  4. sin(900−θ)=cosθsin(900−θ)=cosθ
  5. cos(900−θ)=sinθcos(900−θ)=sinθ
  6. tan(900−θ)=cotθtan(900−θ)=cotθ
  7. sin2θ+cos2θ=1sin2θ+cos2θ=1
  8. sec2θ−tan2θ=1sec2θ−tan2θ=1
  9. tanθ=sinθ/cosθtanθ=sinθ/cosθ
  10. sin(α±β)=sinαcosβ±cosαsinβsin(α±β)=sinαcosβ±cosαsinβ
  11. cos(α±β)=cosαcosβ∓sinαsinβcos(α±β)=cosαcosβ∓sinαsinβ
  12. tan(α±β)=tanα±tanβ1∓tanαtanβtan(α±β)=tanα±tanβ1∓tanαtanβ
  13. sin2θ=2sinθcosθsin2θ=2sinθcosθ
  14. cos2θ=cos2θ−sin2θ=2cos2θ−1=1−2sin2θcos2θ=cos2θ−sin2θ=2cos2θ−1=1−2sin2θ
  15. sinα+sinβ=2sin12(α+β)cos12(α−β)sinα+sinβ=2sin12(α+β)cos12(α−β)
  16. cosα+cosβ=2cos12(α+β)cos12(α−β)cosα+cosβ=2cos12(α+β)cos12(α−β)
  17. s=rθs=rθ

Triangles

  1. Law of sines: asinα=bsinβ=csinγasinα=bsinβ=csinγ
  2. Law of cosines: c2=a2+b2−2abcosγc2=a2+b2−2abcosγ
    Figure shows a triangle with three dissimilar sides labeled a, b and c. All three angles of the triangle are acute angles. The angle between b and c is alpha, the angle between a and c is beta and the angle between a and b is gamma.
  3. Pythagorean theorem: a2+b2=c2a2+b2=c2
    Figure shows a right triangle. Its three sides are labeled a, b and c with c being the hypotenuse. The angle between a and c is labeled theta.

Series expansions

  1. Binomial theorem: (a+b)n=an+nan−1b+n(n−1)an−2b22!+n(n−1)(n−2)an−3b33!+···(a+b)n=an+nan−1b+n(n−1)an−2b22!+n(n−1)(n−2)an−3b33!+···
  2. (1±x)n=1±nx1!+n(n−1)x22!±···(x2<1)(1±x)n=1±nx1!+n(n−1)x22!±···(x2<1)
  3. (1±x)−n=1∓nx1!+n(n+1)x22!∓···(x2<1)(1±x)−n=1∓nx1!+n(n+1)x22!∓···(x2<1)
  4. sinx=x−x33!+x55!−···sinx=x−x33!+x55!−···
  5. cosx=1−x22!+x44!−···cosx=1−x22!+x44!−···
  6. tanx=x+x33+2x515+···tanx=x+x33+2x515+···
  7. ex=1+x+x22!+···ex=1+x+x22!+···
  8. ln(1+x)=x−12x2+13x3−···(|x|<1)ln(1+x)=x−12x2+13x3−···(|x|<1)

Derivatives

  1. ddx[af(x)]=addxf(x)ddx[af(x)]=addxf(x)
  2. ddx[f(x)+g(x)]=ddxf(x)+ddxg(x)ddx[f(x)+g(x)]=ddxf(x)+ddxg(x)
  3. ddx[f(x)g(x)]=f(x)ddxg(x)+g(x)ddxf(x)ddx[f(x)g(x)]=f(x)ddxg(x)+g(x)ddxf(x)
  4. ddxf(u)=[dduf(u)]dudxddxf(u)=[dduf(u)]dudx
  5. ddxxm=mxm−1ddxxm=mxm−1
  6. ddxsinx=cosxddxsinx=cosx
  7. ddxcosx=−sinxddxcosx=−sinx
  8. ddxtanx=sec2xddxtanx=sec2x
  9. ddxcotx=−csc2xddxcotx=−csc2x
  10. ddxsecx=tanxsecxddxsecx=tanxsecx
  11. ddxcscx=−cotxcscxddxcscx=−cotxcscx
  12. ddxex=exddxex=ex
  13. ddxlnx=1xddxlnx=1x
  14. ddxsin−1x=11−x2ddxsin−1x=11−x2
  15. ddxcos−1x=−11−x2ddxcos−1x=−11−x2
  16. ddxtan−1x=11+x2ddxtan−1x=11+x2

Integrals

  1. ∫af(x)dx=a∫f(x)dx∫af(x)dx=a∫f(x)dx
  2. ∫[f(x)+g(x)]dx=∫f(x)dx+∫g(x)dx∫[f(x)+g(x)]dx=∫f(x)dx+∫g(x)dx
  3. ∫xmdx=xm+1m+1(m≠−1)=lnx(m=−1)∫xmdx=xm+1m+1(m≠−1)=lnx(m=−1)
  4. ∫sinxdx=−cosx∫sinxdx=−cosx
  5. ∫cosxdx=sinx∫cosxdx=sinx
  6. ∫tanxdx=ln|secx|∫tanxdx=ln|secx|
  7. ∫sin2axdx=x2−sin2ax4a∫sin2axdx=x2−sin2ax4a
  8. ∫cos2axdx=x2+sin2ax4a∫cos2axdx=x2+sin2ax4a
  9. ∫sinaxcosaxdx=−cos2ax4a∫sinaxcosaxdx=−cos2ax4a
  10. ∫eaxdx=1aeax∫eaxdx=1aeax
  11. ∫xeaxdx=eaxa2(ax−1)∫xeaxdx=eaxa2(ax−1)
  12. ∫lnaxdx=xlnax−x∫lnaxdx=xlnax−x
  13. ∫dxa2+x2=1atan−1xa∫dxa2+x2=1atan−1xa
  14. ∫dxa2−x2=12aln|x+ax−a|∫dxa2−x2=12aln|x+ax−a|
  15. ∫dxa2+x2=sinh−1xa∫dxa2+x2=sinh−1xa
  16. ∫dxa2−x2=sin−1xa∫dxa2−x2=sin−1xa
  17. ∫a2+x2dx=x2a2+x2+a22sinh−1xa∫a2+x2dx=x2a2+x2+a22sinh−1xa
  18. ∫a2−x2dx=x2a2−x2+a22sin−1xa∫a2−x2dx=x2a2−x2+a22sin−1xa
  19. ∫1(x2+a2)3/2dx=xa2x2+a2∫1(x2+a2)3/2dx=xa2x2+a2
  20. ∫x(x2+a2)3/2dx=–1x2+a2∫x(x2+a2)3/2dx=–1x2+a2
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