Skip to Content
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 $=2πr=2πr$ 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

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. $cosα+cosβ=2cos12(α+β)cos12(α−β)cosα+cosβ=2cos12(α+β)cos12(α−β)$

Triangles

1. Law of sines: $asinα=bsinβ=csinγasinα=bsinβ=csinγ$
2. Law of cosines: $c2=a2+b2−2abcosγc2=a2+b2−2abcosγ$
3. Pythagorean theorem: $a2+b2=c2a2+b2=c2$

Series expansions

1. Binomial theorem: $(a+b)n=an+nan−1b+n(n−1)an−2b22!+n(n−1)(n−2)an−3b33!+···(a+b)n=an+nan−1b+n(n−1)an−2b22!+n(n−1)(n−2)an−3b33!+···$
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. $∫a2−x2dx=x2a2−x2+a22sin−1xa∫a2−x2dx=x2a2−x2+a22sin−1xa$
Citation/Attribution

Want to cite, share, or modify this book? This book is Creative Commons Attribution License 4.0 and you must attribute OpenStax.

Attribution information
• If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
Access for free at https://openstax.org/books/university-physics-volume-3/pages/1-introduction
• If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
Access for free at https://openstax.org/books/university-physics-volume-3/pages/1-introduction
Citation information

© Sep 29, 2016 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License 4.0 license. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.