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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±b24ac2ax=b±b24ac2a

Triangle of base bb and height hh Area =12bh=12bh
Circle of radius rr Circumference =2πr=2πr Area =πr2=πr2
Sphere of radius rr Surface area =4πr2=4πr2 Volume =43πr3=43πr3
Cylinder of radius rr and height hh Area of curved surface =2πrh=2πrh Volume =πr2h=πr2h
Table E1 Geometry


Trigonometric Identities

  1. sinθ=1/cscθsinθ=1/cscθ
  2. cosθ=1/secθcosθ=1/secθ
  3. tanθ=1/cotθtanθ=1/cotθ
  4. sin(900θ)=cosθsin(900θ)=cosθ
  5. cos(900θ)=sinθcos(900θ)=sinθ
  6. tan(900θ)=cotθtan(900θ)=cotθ
  7. sin2θ+cos2θ=1sin2θ+cos2θ=1
  8. sec2θtan2θ=1sec2θtan2θ=1
  9. tanθ=sinθ/cosθtanθ=sinθ/cosθ
  10. sin(α±β)=sinαcosβ±cosαsinβsin(α±β)=sinαcosβ±cosαsinβ
  11. cos(α±β)=cosαcosβsinαsinβcos(α±β)=cosαcosβsinαsinβ
  12. tan(α±β)=tanα±tanβ1tanαtanβtan(α±β)=tanα±tanβ1tanαtanβ
  13. sin2θ=2sinθcosθsin2θ=2sinθcosθ
  14. cos2θ=cos2θsin2θ=2cos2θ1=12sin2θcos2θ=cos2θsin2θ=2cos2θ1=12sin2θ
  15. sinα+sinβ=2sin12(α+β)cos12(αβ)sinα+sinβ=2sin12(α+β)cos12(αβ)
  16. cosα+cosβ=2cos12(α+β)cos12(αβ)cosα+cosβ=2cos12(α+β)cos12(αβ)
  17. s=rθs=rθ


  1. Law of sines: asinα=bsinβ=csinγasinα=bsinβ=csinγ
  2. Law of cosines: c2=a2+b22abcosγc2=a2+b22abcosγ
    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+nan1b+n(n1)an2b22!+n(n1)(n2)an3b33!+···(a+b)n=an+nan1b+n(n1)an2b22!+n(n1)(n2)an3b33!+···
  2. (1±x)n=1±nx1!+n(n1)x22!±···(x2<1)(1±x)n=1±nx1!+n(n1)x22!±···(x2<1)
  3. (1±x)n=1nx1!+n(n+1)x22!···(x2<1)(1±x)n=1nx1!+n(n+1)x22!···(x2<1)
  4. sinx=xx33!+x55!···sinx=xx33!+x55!···
  5. cosx=1x22!+x44!···cosx=1x22!+x44!···
  6. tanx=x+x33+2x515+···tanx=x+x33+2x515+···
  7. ex=1+x+x22!+···ex=1+x+x22!+···
  8. ln(1+x)=x12x2+13x3···(|x|<1)ln(1+x)=x12x2+13x3···(|x|<1)


  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=mxm1ddxxm=mxm1
  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=11x2ddxsin−1x=11x2
  15. ddxcos−1x=11x2ddxcos−1x=11x2
  16. ddxtan−1x=11+x2ddxtan−1x=11+x2


  1. af(x)dx=af(x)dxaf(x)dx=af(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(m1)=lnx(m=−1)xmdx=xm+1m+1(m1)=lnx(m=−1)
  4. sinxdx=cosxsinxdx=cosx
  5. cosxdx=sinxcosxdx=sinx
  6. tanxdx=ln|secx|tanxdx=ln|secx|
  7. sin2axdx=x2sin2ax4asin2axdx=x2sin2ax4a
  8. cos2axdx=x2+sin2ax4acos2axdx=x2+sin2ax4a
  9. sinaxcosaxdx=cos2ax4asinaxcosaxdx=cos2ax4a
  10. eaxdx=1aeaxeaxdx=1aeax
  11. xeaxdx=eaxa2(ax1)xeaxdx=eaxa2(ax1)
  12. lnaxdx=xlnaxxlnaxdx=xlnaxx
  13. dxa2+x2=1atan−1xadxa2+x2=1atan−1xa
  14. dxa2x2=12aln|x+axa|dxa2x2=12aln|x+axa|
  15. dxa2+x2=sinh−1xadxa2+x2=sinh−1xa
  16. dxa2x2=sin−1xadxa2x2=sin−1xa
  17. a2+x2dx=x2a2+x2+a22sinh−1xaa2+x2dx=x2a2+x2+a22sinh−1xa
  18. a2x2dx=x2a2x2+a22sin−1xaa2x2dx=x2a2x2+a22sin−1xa
  19. 1(x2+a2)3/2dx=xa2x2+a21(x2+a2)3/2dx=xa2x2+a2
  20. x(x2+a2)3/2dx=1x2+a2x(x2+a2)3/2dx=1x2+a2
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