Appendix E: Mathematical Formulas

Quadratic formula

If ax2+bx+c=0, then x=−b±b2−4ac2a

Trigonometry

Trigonometric Identities

1. sinθ=1/cscθ
2. cosθ=1/secθ
3. tanθ=1/cotθ
4. sin(900−θ)=cosθ
5. cos(900−θ)=sinθ
6. tan(900−θ)=cotθ
7. sin2θ+cos2θ=1
8. sec2θ−tan2θ=1
9. tanθ=sinθ/cosθ
10. sin(α±β)=sinαcosβ±cosαsinβ
11. cos(α±β)=cosαcosβ∓sinαsinβ
12. tan(α±β)=tanα±tanβ1∓tanαtanβ
13. sin2θ=2sinθcosθ
14. cos2θ=cos2θ−sin2θ=2cos2θ−1=1−2sin2θ
15. sinα+sinβ=2sin12(α+β)cos12(α−β)
16. cosα+cosβ=2cos12(α+β)cos12(α−β)

Triangles

1. Law of sines: asinα=bsinβ=csinγ
2. Law of cosines: c2=a2+b2−2abcosγ
   
3. Pythagorean theorem: a2+b2=c2
   

Series expansions

1. Binomial theorem: (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)
3. (1±x)−n=1∓nx1!+n(n+1)x22!∓···(x2<1)
4. sinx=x−x33!+x55!−···
5. cosx=1−x22!+x44!−···
6. tanx=x+x33+2×515+···
7. ex=1+x+x22!+···
8. ln(1+x)=x−12×2+13×3−···(|x|<1)

Derivatives

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

Integrals

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