{"id":263,"date":"2019-04-09T01:59:43","date_gmt":"2019-04-09T05:59:43","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/?post_type=chapter&p=263"},"modified":"2019-04-09T20:57:59","modified_gmt":"2019-04-10T00:57:59","slug":"appendix-e-mathematical-formulas","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/chapter\/appendix-e-mathematical-formulas\/","title":{"raw":"Appendix E: Mathematical Formulas","rendered":"Appendix E: Mathematical Formulas"},"content":{"raw":"

Quadratic formula<\/strong><\/p>\r\n

If\u00a0<\/span>ax2+bx+c=0,<\/span><\/span><\/span>\u00a0<\/span>then\u00a0<\/span>x=\u2212b\u00b1b2\u22124ac2a<\/span><\/span><\/span><\/p>\r\n\r\n

\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Triangle of base\u00a0<\/span>b<\/span><\/span><\/span>\u00a0<\/span>and height\u00a0<\/span>h<\/span><\/span><\/span><\/th>\r\nArea\u00a0<\/span>=12bh<\/span><\/span><\/span><\/th>\r\n<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
Circle of radius\u00a0<\/span>r<\/span><\/span><\/span><\/td>\r\nCircumference\u00a0<\/span>=2\u03c0r<\/span><\/span><\/span><\/td>\r\nArea\u00a0<\/span>=\u03c0r2<\/span><\/span><\/span><\/td>\r\n<\/tr>\r\n
Sphere of radius\u00a0<\/span>r<\/span><\/span><\/span><\/td>\r\nSurface area\u00a0<\/span>=4\u03c0r2<\/span><\/span><\/span><\/td>\r\nVolume\u00a0<\/span>=43\u03c0r3<\/span><\/span><\/span><\/td>\r\n<\/tr>\r\n
Cylinder of radius\u00a0<\/span>r<\/span><\/span><\/span>\u00a0<\/span>and height\u00a0<\/span>h<\/span><\/span><\/span><\/td>\r\nArea of curved surface\u00a0<\/span>=2\u03c0rh<\/span><\/span><\/span><\/td>\r\nVolume\u00a0<\/span>=\u03c0r2h<\/span><\/span><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
Table<\/span>E1<\/span>\u00a0<\/span><\/span>Geometry<\/span><\/span><\/span><\/div>\r\n<\/div>\r\n

Trigonometry<\/strong><\/p>\r\n

Trigonometric Identities<\/em><\/p>\r\n\r\n

    \r\n \t
  1. sin\u03b8=1\/csc\u03b8<\/span><\/span><\/span><\/li>\r\n \t
  2. cos\u03b8=1\/sec\u03b8<\/span><\/span><\/span><\/li>\r\n \t
  3. tan\u03b8=1\/cot\u03b8<\/span><\/span><\/span><\/li>\r\n \t
  4. sin(900\u2212\u03b8)=cos\u03b8<\/span><\/span><\/span><\/li>\r\n \t
  5. cos(900\u2212\u03b8)=sin\u03b8<\/span><\/span><\/span><\/li>\r\n \t
  6. tan(900\u2212\u03b8)=cot\u03b8<\/span><\/span><\/span><\/li>\r\n \t
  7. sin2\u03b8+cos2\u03b8=1<\/span><\/span><\/span><\/li>\r\n \t
  8. sec2\u03b8\u2212tan2\u03b8=1<\/span><\/span><\/span><\/li>\r\n \t
  9. tan\u03b8=sin\u03b8\/cos\u03b8<\/span><\/span><\/span><\/li>\r\n \t
  10. sin(\u03b1\u00b1\u03b2)=sin\u03b1cos\u03b2\u00b1cos\u03b1sin\u03b2<\/span><\/span><\/span><\/li>\r\n \t
  11. cos(\u03b1\u00b1\u03b2)=cos\u03b1cos\u03b2\u2213sin\u03b1sin\u03b2<\/span><\/span><\/span><\/li>\r\n \t
  12. tan(\u03b1\u00b1\u03b2)=tan\u03b1\u00b1tan\u03b21\u2213tan\u03b1tan\u03b2<\/span><\/span><\/span><\/li>\r\n \t
  13. sin2\u03b8=2sin\u03b8cos\u03b8<\/span><\/span><\/span><\/li>\r\n \t
  14. cos2\u03b8=cos2\u03b8\u2212sin2\u03b8=2cos2\u03b8\u22121=1\u22122sin2\u03b8<\/span><\/span><\/span><\/li>\r\n \t
  15. sin\u03b1+sin\u03b2=2sin12(\u03b1+\u03b2)cos12(\u03b1\u2212\u03b2)<\/span><\/span><\/span><\/li>\r\n \t
  16. cos\u03b1+cos\u03b2=2cos12(\u03b1+\u03b2)cos12(\u03b1\u2212\u03b2)<\/span><\/span><\/span><\/li>\r\n<\/ol>\r\n

    Triangles<\/em><\/p>\r\n\r\n

      \r\n \t
    1. Law of sines:\u00a0<\/span>asin\u03b1=bsin\u03b2=csin\u03b3<\/span><\/span><\/span><\/li>\r\n \t
    2. Law of cosines:\u00a0<\/span>c2=a2+b2\u22122abcos\u03b3<\/span><\/span><\/span>\r\n<\/span>\"Figure<\/span><\/li>\r\n \t
    3. Pythagorean theorem:\u00a0<\/span>a2+b2=c2<\/span><\/span><\/span>\r\n<\/span>\"Figure<\/span><\/li>\r\n<\/ol>\r\n

      Series expansions<\/strong><\/p>\r\n\r\n

        \r\n \t
      1. Binomial theorem:\u00a0<\/span>(a+b)n=an+nan\u22121b+n(n\u22121)an\u22122b22!+n(n\u22121)(n\u22122)an\u22123b33!+\u00b7\u00b7\u00b7<\/span><\/span><\/span><\/li>\r\n \t
      2. (1\u00b1x)n=1\u00b1nx1!+n(n\u22121)x22!\u00b1\u00b7\u00b7\u00b7(x2<1)<\/span><\/span><\/span><\/li>\r\n \t
      3. (1\u00b1x)\u2212n=1\u2213nx1!+n(n+1)x22!\u2213\u00b7\u00b7\u00b7(x2<1)<\/span><\/span><\/span><\/li>\r\n \t
      4. sinx=x\u2212x33!+x55!\u2212\u00b7\u00b7\u00b7<\/span><\/span><\/span><\/li>\r\n \t
      5. cosx=1\u2212x22!+x44!\u2212\u00b7\u00b7\u00b7<\/span><\/span><\/span><\/li>\r\n \t
      6. tanx=x+x33+2x515+\u00b7\u00b7\u00b7<\/span><\/span><\/span><\/li>\r\n \t
      7. ex=1+x+x22!+\u00b7\u00b7\u00b7<\/span><\/span><\/span><\/li>\r\n \t
      8. ln(1+x)=x\u221212x2+13x3\u2212\u00b7\u00b7\u00b7(|x|<1)<\/span><\/span><\/span><\/li>\r\n<\/ol>\r\n

        Derivatives<\/strong><\/p>\r\n\r\n

          \r\n \t
        1. ddx[af(x)]=addxf(x)<\/span><\/span><\/span><\/li>\r\n \t
        2. ddx[f(x)+g(x)]=ddxf(x)+ddxg(x)<\/span><\/span><\/span><\/li>\r\n \t
        3. ddx[f(x)g(x)]=f(x)ddxg(x)+g(x)ddxf(x)<\/span><\/span><\/span><\/li>\r\n \t
        4. ddxf(u)=[dduf(u)]dudx<\/span><\/span><\/span><\/li>\r\n \t
        5. ddxxm=mxm\u22121<\/span><\/span><\/span><\/li>\r\n \t
        6. ddxsinx=cosx<\/span><\/span><\/span><\/li>\r\n \t
        7. ddxcosx=\u2212sinx<\/span><\/span><\/span><\/li>\r\n \t
        8. ddxtanx=sec2x<\/span><\/span><\/span><\/li>\r\n \t
        9. ddxcotx=\u2212csc2x<\/span><\/span><\/span><\/li>\r\n \t
        10. ddxsecx=tanxsecx<\/span><\/span><\/span><\/li>\r\n \t
        11. ddxcscx=\u2212cotxcscx<\/span><\/span><\/span><\/li>\r\n \t
        12. ddxex=ex<\/span><\/span><\/span><\/li>\r\n \t
        13. ddxlnx=1x<\/span><\/span><\/span><\/li>\r\n \t
        14. ddxsin\u22121x=11\u2212x2<\/span><\/span><\/span><\/li>\r\n \t
        15. ddxcos\u22121x=\u221211\u2212x2<\/span><\/span><\/span><\/li>\r\n \t
        16. ddxtan\u22121x=\u221211+x2<\/span><\/span><\/span><\/li>\r\n<\/ol>\r\n

          Integrals<\/strong><\/p>\r\n\r\n

            \r\n \t
          1. \u222baf(x)dx=a\u222bf(x)dx<\/span><\/span><\/span><\/li>\r\n \t
          2. \u222b[f(x)+g(x)]dx=\u222bf(x)dx+\u222bg(x)dx<\/span><\/span><\/span><\/li>\r\n \t
          3. \u222bxmdx=xm+1m+1(m\u2260\u22121)=lnx(m=\u22121)<\/span><\/span><\/span><\/li>\r\n \t
          4. \u222bsinxdx=\u2212cosx<\/span><\/span><\/span><\/li>\r\n \t
          5. \u222bcosxdx=sinx<\/span><\/span><\/span><\/li>\r\n \t
          6. \u222btanxdx=ln|secx|<\/span><\/span><\/span><\/li>\r\n \t
          7. \u222bsin2axdx=x2\u2212sin2ax4a<\/span><\/span><\/span><\/li>\r\n \t
          8. \u222bcos2axdx=x2+sin2ax4a<\/span><\/span><\/span><\/li>\r\n \t
          9. \u222bsinaxcosaxdx=\u2212cos2ax4a<\/span><\/span><\/span><\/li>\r\n \t
          10. \u222beaxdx=1aeax<\/span><\/span><\/span><\/li>\r\n \t
          11. \u222bxeaxdx=eaxa2(ax\u22121)<\/span><\/span><\/span><\/li>\r\n \t
          12. \u222blnaxdx=xlnax\u2212x<\/span><\/span><\/span><\/li>\r\n \t
          13. \u222bdxa2+x2=1atan\u22121xa<\/span><\/span><\/span><\/li>\r\n \t
          14. \u222bdxa2\u2212x2=12aln|x+ax\u2212a|<\/span><\/span><\/span><\/li>\r\n \t
          15. \u222bdxa2+x2=sinh\u22121xa<\/span><\/span><\/span><\/li>\r\n \t
          16. \u222bdxa2\u2212x2=sin\u22121xa<\/span><\/span><\/span><\/li>\r\n \t
          17. \u222ba2+x2dx=x2a2+x2+a22sinh\u22121xa<\/span><\/span><\/span><\/li>\r\n \t
          18. \u222ba2\u2212x2dx=x2a2\u2212x2+a22sin\u22121xa<\/span><\/span><\/span><\/li>\r\n<\/ol>\r\n \r\n
            Download for free at http:\/\/cnx.org\/contents\/af275420-6050-4707-995c-57b9cc13c358@11.1<\/em><\/div>","rendered":"

            Quadratic formula<\/strong><\/p>\n

            If\u00a0<\/span>ax2+bx+c=0,<\/span><\/span><\/span>\u00a0<\/span>then\u00a0<\/span>x=\u2212b\u00b1b2\u22124ac2a<\/span><\/span><\/span><\/p>\n

            \n\n\n\n\n\n\n\n
            Triangle of base\u00a0<\/span>b<\/span><\/span><\/span>\u00a0<\/span>and height\u00a0<\/span>h<\/span><\/span><\/span><\/th>\nArea\u00a0<\/span>=12bh<\/span><\/span><\/span><\/th>\n<\/th>\n<\/tr>\n<\/thead>\n
            Circle of radius\u00a0<\/span>r<\/span><\/span><\/span><\/td>\nCircumference\u00a0<\/span>=2\u03c0r<\/span><\/span><\/span><\/td>\nArea\u00a0<\/span>=\u03c0r2<\/span><\/span><\/span><\/td>\n<\/tr>\n
            Sphere of radius\u00a0<\/span>r<\/span><\/span><\/span><\/td>\nSurface area\u00a0<\/span>=4\u03c0r2<\/span><\/span><\/span><\/td>\nVolume\u00a0<\/span>=43\u03c0r3<\/span><\/span><\/span><\/td>\n<\/tr>\n
            Cylinder of radius\u00a0<\/span>r<\/span><\/span><\/span>\u00a0<\/span>and height\u00a0<\/span>h<\/span><\/span><\/span><\/td>\nArea of curved surface\u00a0<\/span>=2\u03c0rh<\/span><\/span><\/span><\/td>\nVolume\u00a0<\/span>=\u03c0r2h<\/span><\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
            Table<\/span>E1<\/span>\u00a0<\/span><\/span>Geometry<\/span><\/span><\/span><\/div>\n<\/div>\n

            Trigonometry<\/strong><\/p>\n

            Trigonometric Identities<\/em><\/p>\n

              \n
            1. sin\u03b8=1\/csc\u03b8<\/span><\/span><\/span><\/li>\n
            2. cos\u03b8=1\/sec\u03b8<\/span><\/span><\/span><\/li>\n
            3. tan\u03b8=1\/cot\u03b8<\/span><\/span><\/span><\/li>\n
            4. sin(900\u2212\u03b8)=cos\u03b8<\/span><\/span><\/span><\/li>\n
            5. cos(900\u2212\u03b8)=sin\u03b8<\/span><\/span><\/span><\/li>\n
            6. tan(900\u2212\u03b8)=cot\u03b8<\/span><\/span><\/span><\/li>\n
            7. sin2\u03b8+cos2\u03b8=1<\/span><\/span><\/span><\/li>\n
            8. sec2\u03b8\u2212tan2\u03b8=1<\/span><\/span><\/span><\/li>\n
            9. tan\u03b8=sin\u03b8\/cos\u03b8<\/span><\/span><\/span><\/li>\n
            10. sin(\u03b1\u00b1\u03b2)=sin\u03b1cos\u03b2\u00b1cos\u03b1sin\u03b2<\/span><\/span><\/span><\/li>\n
            11. cos(\u03b1\u00b1\u03b2)=cos\u03b1cos\u03b2\u2213sin\u03b1sin\u03b2<\/span><\/span><\/span><\/li>\n
            12. tan(\u03b1\u00b1\u03b2)=tan\u03b1\u00b1tan\u03b21\u2213tan\u03b1tan\u03b2<\/span><\/span><\/span><\/li>\n
            13. sin2\u03b8=2sin\u03b8cos\u03b8<\/span><\/span><\/span><\/li>\n
            14. cos2\u03b8=cos2\u03b8\u2212sin2\u03b8=2cos2\u03b8\u22121=1\u22122sin2\u03b8<\/span><\/span><\/span><\/li>\n
            15. sin\u03b1+sin\u03b2=2sin12(\u03b1+\u03b2)cos12(\u03b1\u2212\u03b2)<\/span><\/span><\/span><\/li>\n
            16. cos\u03b1+cos\u03b2=2cos12(\u03b1+\u03b2)cos12(\u03b1\u2212\u03b2)<\/span><\/span><\/span><\/li>\n<\/ol>\n

              Triangles<\/em><\/p>\n

                \n
              1. Law of sines:\u00a0<\/span>asin\u03b1=bsin\u03b2=csin\u03b3<\/span><\/span><\/span><\/li>\n
              2. Law of cosines:\u00a0<\/span>c2=a2+b2\u22122abcos\u03b3<\/span><\/span><\/span>
                \n<\/span>\"Figure<\/span><\/li>\n
              3. Pythagorean theorem:\u00a0<\/span>a2+b2=c2<\/span><\/span><\/span>
                \n<\/span>\"Figure<\/span><\/li>\n<\/ol>\n

                Series expansions<\/strong><\/p>\n

                  \n
                1. Binomial theorem:\u00a0<\/span>(a+b)n=an+nan\u22121b+n(n\u22121)an\u22122b22!+n(n\u22121)(n\u22122)an\u22123b33!+\u00b7\u00b7\u00b7<\/span><\/span><\/span><\/li>\n
                2. (1\u00b1x)n=1\u00b1nx1!+n(n\u22121)x22!\u00b1\u00b7\u00b7\u00b7(x2<1)<\/span><\/span><\/span><\/li>\n
                3. (1\u00b1x)\u2212n=1\u2213nx1!+n(n+1)x22!\u2213\u00b7\u00b7\u00b7(x2<1)<\/span><\/span><\/span><\/li>\n
                4. sinx=x\u2212x33!+x55!\u2212\u00b7\u00b7\u00b7<\/span><\/span><\/span><\/li>\n
                5. cosx=1\u2212x22!+x44!\u2212\u00b7\u00b7\u00b7<\/span><\/span><\/span><\/li>\n
                6. tanx=x+x33+2×515+\u00b7\u00b7\u00b7<\/span><\/span><\/span><\/li>\n
                7. ex=1+x+x22!+\u00b7\u00b7\u00b7<\/span><\/span><\/span><\/li>\n
                8. ln(1+x)=x\u221212×2+13×3\u2212\u00b7\u00b7\u00b7(|x|<1)<\/span><\/span><\/span><\/li>\n<\/ol>\n

                  Derivatives<\/strong><\/p>\n

                    \n
                  1. ddx[af(x)]=addxf(x)<\/span><\/span><\/span><\/li>\n
                  2. ddx[f(x)+g(x)]=ddxf(x)+ddxg(x)<\/span><\/span><\/span><\/li>\n
                  3. ddx[f(x)g(x)]=f(x)ddxg(x)+g(x)ddxf(x)<\/span><\/span><\/span><\/li>\n
                  4. ddxf(u)=[dduf(u)]dudx<\/span><\/span><\/span><\/li>\n
                  5. ddxxm=mxm\u22121<\/span><\/span><\/span><\/li>\n
                  6. ddxsinx=cosx<\/span><\/span><\/span><\/li>\n
                  7. ddxcosx=\u2212sinx<\/span><\/span><\/span><\/li>\n
                  8. ddxtanx=sec2x<\/span><\/span><\/span><\/li>\n
                  9. ddxcotx=\u2212csc2x<\/span><\/span><\/span><\/li>\n
                  10. ddxsecx=tanxsecx<\/span><\/span><\/span><\/li>\n
                  11. ddxcscx=\u2212cotxcscx<\/span><\/span><\/span><\/li>\n
                  12. ddxex=ex<\/span><\/span><\/span><\/li>\n
                  13. ddxlnx=1x<\/span><\/span><\/span><\/li>\n
                  14. ddxsin\u22121x=11\u2212x2<\/span><\/span><\/span><\/li>\n
                  15. ddxcos\u22121x=\u221211\u2212x2<\/span><\/span><\/span><\/li>\n
                  16. ddxtan\u22121x=\u221211+x2<\/span><\/span><\/span><\/li>\n<\/ol>\n

                    Integrals<\/strong><\/p>\n

                      \n
                    1. \u222baf(x)dx=a\u222bf(x)dx<\/span><\/span><\/span><\/li>\n
                    2. \u222b[f(x)+g(x)]dx=\u222bf(x)dx+\u222bg(x)dx<\/span><\/span><\/span><\/li>\n
                    3. \u222bxmdx=xm+1m+1(m\u2260\u22121)=lnx(m=\u22121)<\/span><\/span><\/span><\/li>\n
                    4. \u222bsinxdx=\u2212cosx<\/span><\/span><\/span><\/li>\n
                    5. \u222bcosxdx=sinx<\/span><\/span><\/span><\/li>\n
                    6. \u222btanxdx=ln|secx|<\/span><\/span><\/span><\/li>\n
                    7. \u222bsin2axdx=x2\u2212sin2ax4a<\/span><\/span><\/span><\/li>\n
                    8. \u222bcos2axdx=x2+sin2ax4a<\/span><\/span><\/span><\/li>\n
                    9. \u222bsinaxcosaxdx=\u2212cos2ax4a<\/span><\/span><\/span><\/li>\n
                    10. \u222beaxdx=1aeax<\/span><\/span><\/span><\/li>\n
                    11. \u222bxeaxdx=eaxa2(ax\u22121)<\/span><\/span><\/span><\/li>\n
                    12. \u222blnaxdx=xlnax\u2212x<\/span><\/span><\/span><\/li>\n
                    13. \u222bdxa2+x2=1atan\u22121xa<\/span><\/span><\/span><\/li>\n
                    14. \u222bdxa2\u2212x2=12aln|x+ax\u2212a|<\/span><\/span><\/span><\/li>\n
                    15. \u222bdxa2+x2=sinh\u22121xa<\/span><\/span><\/span><\/li>\n
                    16. \u222bdxa2\u2212x2=sin\u22121xa<\/span><\/span><\/span><\/li>\n
                    17. \u222ba2+x2dx=x2a2+x2+a22sinh\u22121xa<\/span><\/span><\/span><\/li>\n
                    18. \u222ba2\u2212x2dx=x2a2\u2212x2+a22sin\u22121xa<\/span><\/span><\/span><\/li>\n<\/ol>\n

                       <\/p>\n

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