{"id":1054,"date":"2017-11-14T13:47:05","date_gmt":"2017-11-14T13:47:05","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/back-matter\/mathematical-formulas\/"},"modified":"2017-11-14T13:47:05","modified_gmt":"2017-11-14T13:47:05","slug":"mathematical-formulas","status":"publish","type":"back-matter","link":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/back-matter\/mathematical-formulas\/","title":{"raw":"Mathematical Formulas","rendered":"Mathematical Formulas"},"content":{"raw":"\n

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

If [latex]a{x}^{2}+bx+c=0,[\/latex] then [latex]x=\\frac{\\text{\u2212}b\u00b1\\sqrt{{b}^{2}-4ac}}{2a}[\/latex]<\/p>\n\n\n\n\n\n\n\n
Geometry<\/span><\/caption>\n
Triangle of base [latex]b[\/latex] and height [latex]h[\/latex]<\/th>\nArea [latex]=\\frac{1}{2}bh[\/latex]<\/th>\n<\/th>\n<\/tr>\n<\/thead>\n
Circle of radius [latex]r[\/latex]<\/td>\nCircumference [latex]=2\\pi r[\/latex]<\/td>\nArea [latex]=\\pi {r}^{2}[\/latex]<\/td>\n<\/tr>\n
Sphere of radius [latex]r[\/latex]<\/td>\nSurface area [latex]=4\\pi {r}^{2}[\/latex]<\/td>\nVolume [latex]=\\frac{4}{3}\\pi {r}^{3}[\/latex]<\/td>\n<\/tr>\n
Cylinder of radius [latex]r[\/latex] and height [latex]h[\/latex]<\/td>\nArea of curved surface [latex]=2\\pi rh[\/latex]<\/td>\nVolume [latex]=\\pi {r}^{2}h[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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

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

    \n
  1. [latex]\\text{sin}\\phantom{\\rule{0.2em}{0ex}}\\theta =1\\text{\/}\\text{csc}\\phantom{\\rule{0.2em}{0ex}}\\theta [\/latex]<\/li>\n
  2. [latex]\\text{cos}\\phantom{\\rule{0.2em}{0ex}}\\theta =1\\text{\/}\\text{sec}\\phantom{\\rule{0.2em}{0ex}}\\theta [\/latex]<\/li>\n
  3. [latex]\\text{tan}\\phantom{\\rule{0.2em}{0ex}}\\theta =1\\text{\/}\\text{cot}\\phantom{\\rule{0.2em}{0ex}}\\theta [\/latex]<\/li>\n
  4. [latex]\\text{sin}\\left({90}^{0}-\\theta \\right)=\\text{cos}\\phantom{\\rule{0.2em}{0ex}}\\theta [\/latex]<\/li>\n
  5. [latex]\\text{cos}\\left({90}^{0}-\\theta \\right)=\\text{sin}\\phantom{\\rule{0.2em}{0ex}}\\theta [\/latex]<\/li>\n
  6. [latex]\\text{tan}\\left({90}^{0}-\\theta \\right)=\\text{cot}\\phantom{\\rule{0.2em}{0ex}}\\theta [\/latex]<\/li>\n
  7. [latex]{\\text{sin}}^{2}\\phantom{\\rule{0.2em}{0ex}}\\theta +{\\text{cos}}^{2}\\phantom{\\rule{0.2em}{0ex}}\\theta =1[\/latex]<\/li>\n
  8. [latex]{\\text{sec}}^{2}\\phantom{\\rule{0.2em}{0ex}}\\theta -{\\text{tan}}^{2}\\phantom{\\rule{0.2em}{0ex}}\\theta =1[\/latex]<\/li>\n
  9. [latex]\\text{tan}\\phantom{\\rule{0.2em}{0ex}}\\theta =\\text{sin}\\phantom{\\rule{0.2em}{0ex}}\\theta \\text{\/}\\text{cos}\\phantom{\\rule{0.2em}{0ex}}\\theta [\/latex]<\/li>\n
  10. [latex]\\text{sin}\\left(\\alpha \u00b1\\beta \\right)=\\text{sin}\\phantom{\\rule{0.2em}{0ex}}\\alpha \\phantom{\\rule{0.2em}{0ex}}\\text{cos}\\phantom{\\rule{0.2em}{0ex}}\\beta \u00b1\\text{cos}\\phantom{\\rule{0.2em}{0ex}}\\alpha \\phantom{\\rule{0.2em}{0ex}}\\text{sin}\\phantom{\\rule{0.2em}{0ex}}\\beta [\/latex]<\/li>\n
  11. [latex]\\text{cos}\\left(\\alpha \u00b1\\beta \\right)=\\text{cos}\\phantom{\\rule{0.2em}{0ex}}\\alpha \\phantom{\\rule{0.2em}{0ex}}\\text{cos}\\phantom{\\rule{0.2em}{0ex}}\\beta \\mp \\text{sin}\\phantom{\\rule{0.2em}{0ex}}\\alpha \\phantom{\\rule{0.2em}{0ex}}\\text{sin}\\phantom{\\rule{0.2em}{0ex}}\\beta [\/latex]<\/li>\n
  12. [latex]\\text{tan}\\left(\\alpha \u00b1\\beta \\right)=\\frac{\\text{tan}\\phantom{\\rule{0.2em}{0ex}}\\alpha \u00b1\\text{tan}\\phantom{\\rule{0.2em}{0ex}}\\beta }{1\\mp \\text{tan}\\phantom{\\rule{0.2em}{0ex}}\\alpha \\phantom{\\rule{0.2em}{0ex}}\\text{tan}\\phantom{\\rule{0.2em}{0ex}}\\beta }[\/latex]<\/li>\n
  13. [latex]\\text{sin}\\phantom{\\rule{0.2em}{0ex}}2\\theta =2\\text{sin}\\phantom{\\rule{0.2em}{0ex}}\\theta \\phantom{\\rule{0.2em}{0ex}}\\text{cos}\\phantom{\\rule{0.2em}{0ex}}\\theta [\/latex]<\/li>\n
  14. [latex]\\text{cos}\\phantom{\\rule{0.2em}{0ex}}2\\theta ={\\text{cos}}^{2}\\phantom{\\rule{0.2em}{0ex}}\\theta -{\\text{sin}}^{2}\\phantom{\\rule{0.2em}{0ex}}\\theta =2\\phantom{\\rule{0.2em}{0ex}}{\\text{cos}}^{2}\\phantom{\\rule{0.2em}{0ex}}\\theta -1=1-2\\phantom{\\rule{0.2em}{0ex}}{\\text{sin}}^{2}\\phantom{\\rule{0.2em}{0ex}}\\theta [\/latex]<\/li>\n
  15. [latex]\\text{sin}\\phantom{\\rule{0.2em}{0ex}}\\alpha +\\text{sin}\\phantom{\\rule{0.2em}{0ex}}\\beta =2\\phantom{\\rule{0.2em}{0ex}}\\text{sin}\\frac{1}{2}\\left(\\alpha +\\beta \\right)\\text{cos}\\frac{1}{2}\\left(\\alpha -\\beta \\right)[\/latex]<\/li>\n
  16. [latex]\\text{cos}\\phantom{\\rule{0.2em}{0ex}}\\alpha +\\text{cos}\\phantom{\\rule{0.2em}{0ex}}\\beta =2\\phantom{\\rule{0.2em}{0ex}}\\text{cos}\\frac{1}{2}\\left(\\alpha +\\beta \\right)\\text{cos}\\frac{1}{2}\\left(\\alpha -\\beta \\right)[\/latex]<\/li>\n<\/ol>\n

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

      \n
    1. Law of sines: [latex]\\frac{a}{\\text{sin}\\phantom{\\rule{0.2em}{0ex}}\\alpha }=\\frac{b}{\\text{sin}\\phantom{\\rule{0.2em}{0ex}}\\beta }=\\frac{c}{\\text{sin}\\phantom{\\rule{0.2em}{0ex}}\\gamma }[\/latex]<\/li>\n
    2. Law of cosines: [latex]{c}^{2}={a}^{2}+{b}^{2}-2ab\\phantom{\\rule{0.2em}{0ex}}\\text{cos}\\phantom{\\rule{0.2em}{0ex}}\\gamma [\/latex]\n
      <\/div>\n

      \"Figure<\/span><\/p><\/li>\n

    3. Pythagorean theorem: [latex]{a}^{2}+{b}^{2}={c}^{2}[\/latex]\n
      <\/div>\n

      \"Figure<\/span><\/p><\/li>\n<\/ol>\n

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

        \n
      1. Binomial theorem: [latex]{\\left(a+b\\right)}^{n}={a}^{n}+n{a}^{n-1}b+\\frac{n\\left(n-1\\right){a}^{n-2}{b}^{2}}{2\\text{!}}+\\frac{n\\left(n-1\\right)\\left(n-2\\right){a}^{n-3}{b}^{3}}{3\\text{!}}+\\text{\u00b7\u00b7\u00b7}[\/latex]<\/li>\n
      2. [latex]{\\left(1\u00b1x\\right)}^{n}=1\u00b1\\frac{nx}{1\\text{!}}+\\frac{n\\left(n-1\\right){x}^{2}}{2\\text{!}}\u00b1\\text{\u00b7\u00b7\u00b7}\\left({x}^{2}<1\\right)[\/latex]<\/li>\n
      3. [latex]{\\left(1\u00b1x\\right)}^{\\text{\u2212}n}=1\\mp \\frac{nx}{1\\text{!}}+\\frac{n\\left(n+1\\right){x}^{2}}{2\\text{!}}\\mp \\text{\u00b7\u00b7\u00b7}\\left({x}^{2}<1\\right)[\/latex]<\/li>\n
      4. [latex]\\text{sin}\\phantom{\\rule{0.2em}{0ex}}x=x-\\frac{{x}^{3}}{3\\text{!}}+\\frac{{x}^{5}}{5\\text{!}}-\\text{\u00b7\u00b7\u00b7}[\/latex]<\/li>\n
      5. [latex]\\text{cos}\\phantom{\\rule{0.2em}{0ex}}x=1-\\frac{{x}^{2}}{2\\text{!}}+\\frac{{x}^{4}}{4\\text{!}}-\\text{\u00b7\u00b7\u00b7}[\/latex]<\/li>\n
      6. [latex]\\text{tan}\\phantom{\\rule{0.2em}{0ex}}x=x+\\frac{{x}^{3}}{3}+\\frac{2{x}^{5}}{15}+\\text{\u00b7\u00b7\u00b7}[\/latex]<\/li>\n
      7. [latex]{e}^{x}=1+x+\\frac{{x}^{2}}{2\\text{!}}+\\text{\u00b7\u00b7\u00b7}[\/latex]<\/li>\n
      8. [latex]\\text{ln}\\left(1+x\\right)=x-\\frac{1}{2}{x}^{2}+\\frac{1}{3}{x}^{3}-\\text{\u00b7\u00b7\u00b7}\\left(|x|<1\\right)[\/latex]<\/li>\n<\/ol>\n

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

          \n
        1. [latex]\\frac{d}{dx}\\left[af\\left(x\\right)\\right]=a\\frac{d}{dx}f\\left(x\\right)[\/latex]<\/li>\n
        2. [latex]\\frac{d}{dx}\\left[f\\left(x\\right)+g\\left(x\\right)\\right]=\\frac{d}{dx}f\\left(x\\right)+\\frac{d}{dx}g\\left(x\\right)[\/latex]<\/li>\n
        3. [latex]\\frac{d}{dx}\\left[f\\left(x\\right)g\\left(x\\right)\\right]=f\\left(x\\right)\\frac{d}{dx}g\\left(x\\right)+g\\left(x\\right)\\frac{d}{dx}f\\left(x\\right)[\/latex]<\/li>\n
        4. [latex]\\frac{d}{dx}f\\left(u\\right)=\\left[\\frac{d}{du}f\\left(u\\right)\\right]\\frac{du}{dx}[\/latex]<\/li>\n
        5. [latex]\\frac{d}{dx}{x}^{m}=m{x}^{m-1}[\/latex]<\/li>\n
        6. [latex]\\frac{d}{dx}\\phantom{\\rule{0.2em}{0ex}}\\text{sin}\\phantom{\\rule{0.2em}{0ex}}x=\\text{cos}\\phantom{\\rule{0.2em}{0ex}}x[\/latex]<\/li>\n
        7. [latex]\\frac{d}{dx}\\phantom{\\rule{0.2em}{0ex}}\\text{cos}\\phantom{\\rule{0.2em}{0ex}}x=\\text{\u2212}\\text{sin}\\phantom{\\rule{0.2em}{0ex}}x[\/latex]<\/li>\n
        8. [latex]\\frac{d}{dx}\\phantom{\\rule{0.2em}{0ex}}\\text{tan}\\phantom{\\rule{0.2em}{0ex}}x={\\text{sec}}^{2}\\phantom{\\rule{0.2em}{0ex}}x[\/latex]<\/li>\n
        9. [latex]\\frac{d}{dx}\\phantom{\\rule{0.2em}{0ex}}\\text{cot}\\phantom{\\rule{0.2em}{0ex}}x=\\text{\u2212}{\\text{csc}}^{2}\\phantom{\\rule{0.2em}{0ex}}x[\/latex]<\/li>\n
        10. [latex]\\frac{d}{dx}\\phantom{\\rule{0.2em}{0ex}}\\text{sec}\\phantom{\\rule{0.2em}{0ex}}x=\\text{tan}\\phantom{\\rule{0.2em}{0ex}}x\\phantom{\\rule{0.2em}{0ex}}\\text{sec}\\phantom{\\rule{0.2em}{0ex}}x[\/latex]<\/li>\n
        11. [latex]\\frac{d}{dx}\\phantom{\\rule{0.2em}{0ex}}\\text{csc}\\phantom{\\rule{0.2em}{0ex}}x=\\text{\u2212}\\text{cot}\\phantom{\\rule{0.2em}{0ex}}x\\phantom{\\rule{0.2em}{0ex}}\\text{csc}\\phantom{\\rule{0.2em}{0ex}}x[\/latex]<\/li>\n
        12. [latex]\\frac{d}{dx}{e}^{x}={e}^{x}[\/latex]<\/li>\n
        13. [latex]\\frac{d}{dx}\\phantom{\\rule{0.2em}{0ex}}\\text{ln}\\phantom{\\rule{0.2em}{0ex}}x=\\frac{1}{x}[\/latex]<\/li>\n
        14. [latex]\\frac{d}{dx}\\phantom{\\rule{0.2em}{0ex}}{\\text{sin}}^{-1}\\phantom{\\rule{0.2em}{0ex}}x=\\frac{1}{\\sqrt{1-{x}^{2}}}[\/latex]<\/li>\n
        15. [latex]\\frac{d}{dx}\\phantom{\\rule{0.2em}{0ex}}{\\text{cos}}^{-1}x=-\\frac{1}{\\sqrt{1-{x}^{2}}}[\/latex]<\/li>\n
        16. [latex]\\frac{d}{dx}\\phantom{\\rule{0.2em}{0ex}}{\\text{tan}}^{-1}x=-\\frac{1}{1+{x}^{2}}[\/latex]<\/li>\n<\/ol>\n

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

            \n
          1. [latex]\\int af\\left(x\\right)dx=a\\int f\\left(x\\right)dx[\/latex]<\/li>\n
          2. [latex]\\int \\left[f\\left(x\\right)+g\\left(x\\right)\\right]dx=\\int f\\left(x\\right)dx+\\int g\\left(x\\right)dx[\/latex]<\/li>\n
          3. [latex]\\begin{array}{cc}\\hfill \\int {x}^{m}dx& =\\frac{{x}^{m+1}}{m+1}\\phantom{\\rule{0.2em}{0ex}}\\left(m\\ne \\text{\u2212}1\\right)\\hfill \\\\ & =\\text{ln}\\phantom{\\rule{0.2em}{0ex}}x\\left(m=-1\\right)\\hfill \\end{array}[\/latex]<\/li>\n
          4. [latex]\\int \\text{sin}\\phantom{\\rule{0.2em}{0ex}}x\\phantom{\\rule{0.2em}{0ex}}dx=\\text{\u2212}\\text{cos}\\phantom{\\rule{0.2em}{0ex}}x[\/latex]<\/li>\n
          5. [latex]\\int \\text{cos}\\phantom{\\rule{0.2em}{0ex}}x\\phantom{\\rule{0.2em}{0ex}}dx=\\text{sin}\\phantom{\\rule{0.2em}{0ex}}x[\/latex]<\/li>\n
          6. [latex]\\int \\text{tan}\\phantom{\\rule{0.2em}{0ex}}x\\phantom{\\rule{0.2em}{0ex}}dx=\\text{ln}|\\text{sec}\\phantom{\\rule{0.2em}{0ex}}x|[\/latex]<\/li>\n
          7. [latex]\\int {\\text{sin}}^{2}\\phantom{\\rule{0.2em}{0ex}}ax\\phantom{\\rule{0.2em}{0ex}}dx=\\frac{x}{2}-\\frac{\\text{sin}\\phantom{\\rule{0.2em}{0ex}}2ax}{4a}[\/latex]<\/li>\n
          8. [latex]\\int {\\text{cos}}^{2}\\phantom{\\rule{0.2em}{0ex}}ax\\phantom{\\rule{0.2em}{0ex}}dx=\\frac{x}{2}+\\frac{\\text{sin}\\phantom{\\rule{0.2em}{0ex}}2ax}{4a}[\/latex]<\/li>\n
          9. [latex]\\int \\text{sin}\\phantom{\\rule{0.2em}{0ex}}ax\\phantom{\\rule{0.2em}{0ex}}\\text{cos}\\phantom{\\rule{0.2em}{0ex}}ax\\phantom{\\rule{0.2em}{0ex}}dx=-\\frac{\\text{cos}2ax}{4a}[\/latex]<\/li>\n
          10. [latex]\\int {e}^{ax}\\phantom{\\rule{0.2em}{0ex}}dx=\\frac{1}{a}{e}^{ax}[\/latex]<\/li>\n
          11. [latex]\\int x{e}^{ax}dx=\\frac{{e}^{ax}}{{a}^{2}}\\left(ax-1\\right)[\/latex]<\/li>\n
          12. [latex]\\int \\text{ln}\\phantom{\\rule{0.2em}{0ex}}ax\\phantom{\\rule{0.2em}{0ex}}dx=x\\phantom{\\rule{0.2em}{0ex}}\\text{ln}\\phantom{\\rule{0.2em}{0ex}}ax-x[\/latex]<\/li>\n
          13. [latex]\\int \\frac{dx}{{a}^{2}+{x}^{2}}=\\frac{1}{a}\\phantom{\\rule{0.2em}{0ex}}{\\text{tan}}^{-1}\\frac{x}{a}[\/latex]<\/li>\n
          14. [latex]\\int \\frac{dx}{{a}^{2}-{x}^{2}}=\\frac{1}{2a}\\phantom{\\rule{0.2em}{0ex}}\\text{ln}|\\frac{x+a}{x-a}|[\/latex]<\/li>\n
          15. [latex]\\int \\frac{dx}{\\sqrt{{a}^{2}+{x}^{2}}}={\\text{sinh}}^{-1}\\frac{x}{a}[\/latex]<\/li>\n
          16. [latex]\\int \\frac{dx}{\\sqrt{{a}^{2}-{x}^{2}}}={\\text{sin}}^{-1}\\frac{x}{a}[\/latex]<\/li>\n
          17. [latex]\\int \\sqrt{{a}^{2}+{x}^{2}}\\phantom{\\rule{0.2em}{0ex}}dx=\\frac{x}{2}\\sqrt{{a}^{2}+{x}^{2}}+\\frac{{a}^{2}}{2}\\phantom{\\rule{0.2em}{0ex}}{\\text{sinh}}^{-1}\\frac{x}{a}[\/latex]<\/li>\n
          18. [latex]\\int \\sqrt{{a}^{2}-{x}^{2}}\\phantom{\\rule{0.2em}{0ex}}dx=\\frac{x}{2}\\sqrt{{a}^{2}-{x}^{2}}+\\frac{{a}^{2}}{2}\\phantom{\\rule{0.2em}{0ex}}{\\text{sin}}^{-1}\\frac{x}{a}[\/latex]<\/li>\n<\/ol>\n\n","rendered":"

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

            If \"a{x}^{2}+bx+c=0,\" then \"x=\frac{\text{−}b±\sqrt{{b}^{2}-4ac}}{2a}\"<\/p>\n\n\n\n\n\n\n\n
            Geometry<\/span><\/caption>\n
            Triangle of base \"b\" and height \"h\"<\/th>\nArea \"=\frac{1}{2}bh\"<\/th>\n<\/th>\n<\/tr>\n<\/thead>\n
            Circle of radius \"r\"<\/td>\nCircumference \"=2\pi r\"<\/td>\nArea \"=\pi {r}^{2}\"<\/td>\n<\/tr>\n
            Sphere of radius \"r\"<\/td>\nSurface area \"=4\pi {r}^{2}\"<\/td>\nVolume \"=\frac{4}{3}\pi {r}^{3}\"<\/td>\n<\/tr>\n
            Cylinder of radius \"r\" and height \"h\"<\/td>\nArea of curved surface \"=2\pi rh\"<\/td>\nVolume \"=\pi {r}^{2}h\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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

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

              \n
            1. \"\text{sin}\phantom{\rule{0.2em}{0ex}}\theta =1\text{/}\text{csc}\phantom{\rule{0.2em}{0ex}}\theta \"<\/li>\n
            2. \"\text{cos}\phantom{\rule{0.2em}{0ex}}\theta =1\text{/}\text{sec}\phantom{\rule{0.2em}{0ex}}\theta \"<\/li>\n
            3. \"\text{tan}\phantom{\rule{0.2em}{0ex}}\theta =1\text{/}\text{cot}\phantom{\rule{0.2em}{0ex}}\theta \"<\/li>\n
            4. \"\text{sin}\left({90}^{0}-\theta \right)=\text{cos}\phantom{\rule{0.2em}{0ex}}\theta \"<\/li>\n
            5. \"\text{cos}\left({90}^{0}-\theta \right)=\text{sin}\phantom{\rule{0.2em}{0ex}}\theta \"<\/li>\n
            6. \"\text{tan}\left({90}^{0}-\theta \right)=\text{cot}\phantom{\rule{0.2em}{0ex}}\theta \"<\/li>\n
            7. \"{\text{sin}}^{2}\phantom{\rule{0.2em}{0ex}}\theta +{\text{cos}}^{2}\phantom{\rule{0.2em}{0ex}}\theta =1\"<\/li>\n
            8. \"{\text{sec}}^{2}\phantom{\rule{0.2em}{0ex}}\theta -{\text{tan}}^{2}\phantom{\rule{0.2em}{0ex}}\theta =1\"<\/li>\n
            9. \"\text{tan}\phantom{\rule{0.2em}{0ex}}\theta =\text{sin}\phantom{\rule{0.2em}{0ex}}\theta \text{/}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta \"<\/li>\n
            10. \"\text{sin}\left(\alpha ±\beta \right)=\text{sin}\phantom{\rule{0.2em}{0ex}}\alpha \phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\beta ±\text{cos}\phantom{\rule{0.2em}{0ex}}\alpha \phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\beta \"<\/li>\n
            11. \"\text{cos}\left(\alpha ±\beta \right)=\text{cos}\phantom{\rule{0.2em}{0ex}}\alpha \phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\beta \mp \text{sin}\phantom{\rule{0.2em}{0ex}}\alpha \phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\beta \"<\/li>\n
            12. \"\text{tan}\left(\alpha ±\beta \right)=\frac{\text{tan}\phantom{\rule{0.2em}{0ex}}\alpha ±\text{tan}\phantom{\rule{0.2em}{0ex}}\beta }{1\mp \text{tan}\phantom{\rule{0.2em}{0ex}}\alpha \phantom{\rule{0.2em}{0ex}}\text{tan}\phantom{\rule{0.2em}{0ex}}\beta }\"<\/li>\n
            13. \"\text{sin}\phantom{\rule{0.2em}{0ex}}2\theta =2\text{sin}\phantom{\rule{0.2em}{0ex}}\theta \phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta \"<\/li>\n
            14. \"\text{cos}\phantom{\rule{0.2em}{0ex}}2\theta ={\text{cos}}^{2}\phantom{\rule{0.2em}{0ex}}\theta -{\text{sin}}^{2}\phantom{\rule{0.2em}{0ex}}\theta =2\phantom{\rule{0.2em}{0ex}}{\text{cos}}^{2}\phantom{\rule{0.2em}{0ex}}\theta -1=1-2\phantom{\rule{0.2em}{0ex}}{\text{sin}}^{2}\phantom{\rule{0.2em}{0ex}}\theta \"<\/li>\n
            15. \"\text{sin}\phantom{\rule{0.2em}{0ex}}\alpha +\text{sin}\phantom{\rule{0.2em}{0ex}}\beta =2\phantom{\rule{0.2em}{0ex}}\text{sin}\frac{1}{2}\left(\alpha +\beta \right)\text{cos}\frac{1}{2}\left(\alpha -\beta \right)\"<\/li>\n
            16. \"\text{cos}\phantom{\rule{0.2em}{0ex}}\alpha +\text{cos}\phantom{\rule{0.2em}{0ex}}\beta =2\phantom{\rule{0.2em}{0ex}}\text{cos}\frac{1}{2}\left(\alpha +\beta \right)\text{cos}\frac{1}{2}\left(\alpha -\beta \right)\"<\/li>\n<\/ol>\n

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

                \n
              1. Law of sines: \"\frac{a}{\text{sin}\phantom{\rule{0.2em}{0ex}}\alpha }=\frac{b}{\text{sin}\phantom{\rule{0.2em}{0ex}}\beta }=\frac{c}{\text{sin}\phantom{\rule{0.2em}{0ex}}\gamma }\"<\/li>\n
              2. Law of cosines: \"{c}^{2}={a}^{2}+{b}^{2}-2ab\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\gamma \"\n
                <\/div>\n

                \"Figure<\/span><\/p>\n<\/li>\n

              3. Pythagorean theorem: \"{a}^{2}+{b}^{2}={c}^{2}\"\n
                <\/div>\n

                \"Figure<\/span><\/p>\n<\/li>\n<\/ol>\n

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

                  \n
                1. Binomial theorem: \"{\left(a+b\right)}^{n}={a}^{n}+n{a}^{n-1}b+\frac{n\left(n-1\right){a}^{n-2}{b}^{2}}{2\text{!}}+\frac{n\left(n-1\right)\left(n-2\right){a}^{n-3}{b}^{3}}{3\text{!}}+\text{···}\"<\/li>\n
                2. \"{\left(1±x\right)}^{n}=1±\frac{nx}{1\text{!}}+\frac{n\left(n-1\right){x}^{2}}{2\text{!}}±\text{···}\left({x}^{2}<1\right)\"<\/li>\n
                3. \"{\left(1±x\right)}^{\text{−}n}=1\mp \frac{nx}{1\text{!}}+\frac{n\left(n+1\right){x}^{2}}{2\text{!}}\mp \text{···}\left({x}^{2}<1\right)\"<\/li>\n
                4. \"\text{sin}\phantom{\rule{0.2em}{0ex}}x=x-\frac{{x}^{3}}{3\text{!}}+\frac{{x}^{5}}{5\text{!}}-\text{···}\"<\/li>\n
                5. \"\text{cos}\phantom{\rule{0.2em}{0ex}}x=1-\frac{{x}^{2}}{2\text{!}}+\frac{{x}^{4}}{4\text{!}}-\text{···}\"<\/li>\n
                6. \"\text{tan}\phantom{\rule{0.2em}{0ex}}x=x+\frac{{x}^{3}}{3}+\frac{2{x}^{5}}{15}+\text{···}\"<\/li>\n
                7. \"{e}^{x}=1+x+\frac{{x}^{2}}{2\text{!}}+\text{···}\"<\/li>\n
                8. \"\text{ln}\left(1+x\right)=x-\frac{1}{2}{x}^{2}+\frac{1}{3}{x}^{3}-\text{···}\left(|x|<1\right)\"<\/li>\n<\/ol>\n

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

                    \n
                  1. \"\frac{d}{dx}\left[af\left(x\right)\right]=a\frac{d}{dx}f\left(x\right)\"<\/li>\n
                  2. \"\frac{d}{dx}\left[f\left(x\right)+g\left(x\right)\right]=\frac{d}{dx}f\left(x\right)+\frac{d}{dx}g\left(x\right)\"<\/li>\n
                  3. \"\frac{d}{dx}\left[f\left(x\right)g\left(x\right)\right]=f\left(x\right)\frac{d}{dx}g\left(x\right)+g\left(x\right)\frac{d}{dx}f\left(x\right)\"<\/li>\n
                  4. \"\frac{d}{dx}f\left(u\right)=\left[\frac{d}{du}f\left(u\right)\right]\frac{du}{dx}\"<\/li>\n
                  5. \"\frac{d}{dx}{x}^{m}=m{x}^{m-1}\"<\/li>\n
                  6. \"\frac{d}{dx}\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}x=\text{cos}\phantom{\rule{0.2em}{0ex}}x\"<\/li>\n
                  7. \"\frac{d}{dx}\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}x=\text{−}\text{sin}\phantom{\rule{0.2em}{0ex}}x\"<\/li>\n
                  8. \"\frac{d}{dx}\phantom{\rule{0.2em}{0ex}}\text{tan}\phantom{\rule{0.2em}{0ex}}x={\text{sec}}^{2}\phantom{\rule{0.2em}{0ex}}x\"<\/li>\n
                  9. \"\frac{d}{dx}\phantom{\rule{0.2em}{0ex}}\text{cot}\phantom{\rule{0.2em}{0ex}}x=\text{−}{\text{csc}}^{2}\phantom{\rule{0.2em}{0ex}}x\"<\/li>\n
                  10. \"\frac{d}{dx}\phantom{\rule{0.2em}{0ex}}\text{sec}\phantom{\rule{0.2em}{0ex}}x=\text{tan}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{sec}\phantom{\rule{0.2em}{0ex}}x\"<\/li>\n
                  11. \"\frac{d}{dx}\phantom{\rule{0.2em}{0ex}}\text{csc}\phantom{\rule{0.2em}{0ex}}x=\text{−}\text{cot}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{csc}\phantom{\rule{0.2em}{0ex}}x\"<\/li>\n
                  12. \"\frac{d}{dx}{e}^{x}={e}^{x}\"<\/li>\n
                  13. \"\frac{d}{dx}\phantom{\rule{0.2em}{0ex}}\text{ln}\phantom{\rule{0.2em}{0ex}}x=\frac{1}{x}\"<\/li>\n
                  14. \"\frac{d}{dx}\phantom{\rule{0.2em}{0ex}}{\text{sin}}^{-1}\phantom{\rule{0.2em}{0ex}}x=\frac{1}{\sqrt{1-{x}^{2}}}\"<\/li>\n
                  15. \"\frac{d}{dx}\phantom{\rule{0.2em}{0ex}}{\text{cos}}^{-1}x=-\frac{1}{\sqrt{1-{x}^{2}}}\"<\/li>\n
                  16. \"\frac{d}{dx}\phantom{\rule{0.2em}{0ex}}{\text{tan}}^{-1}x=-\frac{1}{1+{x}^{2}}\"<\/li>\n<\/ol>\n

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

                      \n
                    1. \"\int af\left(x\right)dx=a\int f\left(x\right)dx\"<\/li>\n
                    2. \"\int \left[f\left(x\right)+g\left(x\right)\right]dx=\int f\left(x\right)dx+\int g\left(x\right)dx\"<\/li>\n
                    3. \"\begin{array}{cc}\hfill \int {x}^{m}dx& =\frac{{x}^{m+1}}{m+1}\phantom{\rule{0.2em}{0ex}}\left(m\ne \text{−}1\right)\hfill \\ & =\text{ln}\phantom{\rule{0.2em}{0ex}}x\left(m=-1\right)\hfill \end{array}\"<\/li>\n
                    4. \"\int \text{sin}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}dx=\text{−}\text{cos}\phantom{\rule{0.2em}{0ex}}x\"<\/li>\n
                    5. \"\int \text{cos}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}dx=\text{sin}\phantom{\rule{0.2em}{0ex}}x\"<\/li>\n
                    6. \"\int \text{tan}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}dx=\text{ln}|\text{sec}\phantom{\rule{0.2em}{0ex}}x|\"<\/li>\n
                    7. \"\int {\text{sin}}^{2}\phantom{\rule{0.2em}{0ex}}ax\phantom{\rule{0.2em}{0ex}}dx=\frac{x}{2}-\frac{\text{sin}\phantom{\rule{0.2em}{0ex}}2ax}{4a}\"<\/li>\n
                    8. \"\int {\text{cos}}^{2}\phantom{\rule{0.2em}{0ex}}ax\phantom{\rule{0.2em}{0ex}}dx=\frac{x}{2}+\frac{\text{sin}\phantom{\rule{0.2em}{0ex}}2ax}{4a}\"<\/li>\n
                    9. \"\int \text{sin}\phantom{\rule{0.2em}{0ex}}ax\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}ax\phantom{\rule{0.2em}{0ex}}dx=-\frac{\text{cos}2ax}{4a}\"<\/li>\n
                    10. \"\int {e}^{ax}\phantom{\rule{0.2em}{0ex}}dx=\frac{1}{a}{e}^{ax}\"<\/li>\n
                    11. \"\int x{e}^{ax}dx=\frac{{e}^{ax}}{{a}^{2}}\left(ax-1\right)\"<\/li>\n
                    12. \"\int \text{ln}\phantom{\rule{0.2em}{0ex}}ax\phantom{\rule{0.2em}{0ex}}dx=x\phantom{\rule{0.2em}{0ex}}\text{ln}\phantom{\rule{0.2em}{0ex}}ax-x\"<\/li>\n
                    13. \"\int \frac{dx}{{a}^{2}+{x}^{2}}=\frac{1}{a}\phantom{\rule{0.2em}{0ex}}{\text{tan}}^{-1}\frac{x}{a}\"<\/li>\n
                    14. \"\int \frac{dx}{{a}^{2}-{x}^{2}}=\frac{1}{2a}\phantom{\rule{0.2em}{0ex}}\text{ln}|\frac{x+a}{x-a}|\"<\/li>\n
                    15. \"\int \frac{dx}{\sqrt{{a}^{2}+{x}^{2}}}={\text{sinh}}^{-1}\frac{x}{a}\"<\/li>\n
                    16. \"\int \frac{dx}{\sqrt{{a}^{2}-{x}^{2}}}={\text{sin}}^{-1}\frac{x}{a}\"<\/li>\n
                    17. \"\int \sqrt{{a}^{2}+{x}^{2}}\phantom{\rule{0.2em}{0ex}}dx=\frac{x}{2}\sqrt{{a}^{2}+{x}^{2}}+\frac{{a}^{2}}{2}\phantom{\rule{0.2em}{0ex}}{\text{sinh}}^{-1}\frac{x}{a}\"<\/li>\n
                    18. \"\int \sqrt{{a}^{2}-{x}^{2}}\phantom{\rule{0.2em}{0ex}}dx=\frac{x}{2}\sqrt{{a}^{2}-{x}^{2}}+\frac{{a}^{2}}{2}\phantom{\rule{0.2em}{0ex}}{\text{sin}}^{-1}\frac{x}{a}\"<\/li>\n<\/ol>\n","protected":false},"author":211,"menu_order":6,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":"all-rights-reserved"},"back-matter-type":[],"contributor":[],"license":[56],"class_list":["post-1054","back-matter","type-back-matter","status-publish","hentry","license-all-rights-reserved"],"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-json\/pressbooks\/v2\/back-matter\/1054","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-json\/pressbooks\/v2\/back-matter"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-json\/wp\/v2\/types\/back-matter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-json\/wp\/v2\/users\/211"}],"version-history":[{"count":0,"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-json\/pressbooks\/v2\/back-matter\/1054\/revisions"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-json\/pressbooks\/v2\/back-matter\/1054\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-json\/wp\/v2\/media?parent=1054"}],"wp:term":[{"taxonomy":"back-matter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-json\/pressbooks\/v2\/back-matter-type?post=1054"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-json\/wp\/v2\/contributor?post=1054"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandboxbook1\/wp-json\/wp\/v2\/license?post=1054"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}