{"id":27,"date":"2017-11-14T13:47:05","date_gmt":"2017-11-14T18:47:05","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/back-matter\/mathematical-formulas\/"},"modified":"2017-11-14T13:47:05","modified_gmt":"2017-11-14T18:47:05","slug":"mathematical-formulas","status":"publish","type":"back-matter","link":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/back-matter\/mathematical-formulas\/","title":{"raw":"Mathematical Formulas","rendered":"Mathematical Formulas"},"content":{"raw":"

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

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

Trigonometry<\/strong><\/p>

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

  1. [latex]\\text{sin}\\phantom{\\rule{0.2em}{0ex}}\\theta =1\\text{\/}\\text{csc}\\phantom{\\rule{0.2em}{0ex}}\\theta [\/latex]<\/li>
  2. [latex]\\text{cos}\\phantom{\\rule{0.2em}{0ex}}\\theta =1\\text{\/}\\text{sec}\\phantom{\\rule{0.2em}{0ex}}\\theta [\/latex]<\/li>
  3. [latex]\\text{tan}\\phantom{\\rule{0.2em}{0ex}}\\theta =1\\text{\/}\\text{cot}\\phantom{\\rule{0.2em}{0ex}}\\theta [\/latex]<\/li>
  4. [latex]\\text{sin}\\left({90}^{0}-\\theta \\right)=\\text{cos}\\phantom{\\rule{0.2em}{0ex}}\\theta [\/latex]<\/li>
  5. [latex]\\text{cos}\\left({90}^{0}-\\theta \\right)=\\text{sin}\\phantom{\\rule{0.2em}{0ex}}\\theta [\/latex]<\/li>
  6. [latex]\\text{tan}\\left({90}^{0}-\\theta \\right)=\\text{cot}\\phantom{\\rule{0.2em}{0ex}}\\theta [\/latex]<\/li>
  7. [latex]{\\text{sin}}^{2}\\phantom{\\rule{0.2em}{0ex}}\\theta +{\\text{cos}}^{2}\\phantom{\\rule{0.2em}{0ex}}\\theta =1[\/latex]<\/li>
  8. [latex]{\\text{sec}}^{2}\\phantom{\\rule{0.2em}{0ex}}\\theta -{\\text{tan}}^{2}\\phantom{\\rule{0.2em}{0ex}}\\theta =1[\/latex]<\/li>
  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>
  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>
  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>
  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>
  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>
  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>
  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>
  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><\/ol>

    Triangles<\/em><\/p>

    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>
    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]

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

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

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

      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>
      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>
      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>
      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>
      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>
      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>
      7. [latex]{e}^{x}=1+x+\\frac{{x}^{2}}{2\\text{!}}+\\text{\u00b7\u00b7\u00b7}[\/latex]<\/li>
      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><\/ol>

        Derivatives<\/strong><\/p>

        1. [latex]\\frac{d}{dx}\\left[af\\left(x\\right)\\right]=a\\frac{d}{dx}f\\left(x\\right)[\/latex]<\/li>
        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>
        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>
        4. [latex]\\frac{d}{dx}f\\left(u\\right)=\\left[\\frac{d}{du}f\\left(u\\right)\\right]\\frac{du}{dx}[\/latex]<\/li>
        5. [latex]\\frac{d}{dx}{x}^{m}=m{x}^{m-1}[\/latex]<\/li>
        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>
        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>
        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>
        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>
        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>
        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>
        12. [latex]\\frac{d}{dx}{e}^{x}={e}^{x}[\/latex]<\/li>
        13. [latex]\\frac{d}{dx}\\phantom{\\rule{0.2em}{0ex}}\\text{ln}\\phantom{\\rule{0.2em}{0ex}}x=\\frac{1}{x}[\/latex]<\/li>
        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>
        15. [latex]\\frac{d}{dx}\\phantom{\\rule{0.2em}{0ex}}{\\text{cos}}^{-1}x=-\\frac{1}{\\sqrt{1-{x}^{2}}}[\/latex]<\/li>
        16. [latex]\\frac{d}{dx}\\phantom{\\rule{0.2em}{0ex}}{\\text{tan}}^{-1}x=-\\frac{1}{1+{x}^{2}}[\/latex]<\/li><\/ol>

          Integrals<\/strong><\/p>

          1. [latex]\\int af\\left(x\\right)dx=a\\int f\\left(x\\right)dx[\/latex]<\/li>
          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>
          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>
          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>
          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>
          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>
          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>
          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>
          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>
          10. [latex]\\int {e}^{ax}\\phantom{\\rule{0.2em}{0ex}}dx=\\frac{1}{a}{e}^{ax}[\/latex]<\/li>
          11. [latex]\\int x{e}^{ax}dx=\\frac{{e}^{ax}}{{a}^{2}}\\left(ax-1\\right)[\/latex]<\/li>
          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>
          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>
          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>
          15. [latex]\\int \\frac{dx}{\\sqrt{{a}^{2}+{x}^{2}}}={\\text{sinh}}^{-1}\\frac{x}{a}[\/latex]<\/li>
          16. [latex]\\int \\frac{dx}{\\sqrt{{a}^{2}-{x}^{2}}}={\\text{sin}}^{-1}\\frac{x}{a}[\/latex]<\/li>
          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>
          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><\/ol>","rendered":"

            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><\/li>\n

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

                \"Figure<\/span><\/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","protected":false},"author":211,"menu_order":6,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"back-matter-type":[],"contributor":[],"license":[],"class_list":["post-27","back-matter","type-back-matter","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-json\/pressbooks\/v2\/back-matter\/27","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-json\/pressbooks\/v2\/back-matter"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-json\/wp\/v2\/types\/back-matter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-json\/wp\/v2\/users\/211"}],"version-history":[{"count":0,"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-json\/pressbooks\/v2\/back-matter\/27\/revisions"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-json\/pressbooks\/v2\/back-matter\/27\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-json\/wp\/v2\/media?parent=27"}],"wp:term":[{"taxonomy":"back-matter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-json\/pressbooks\/v2\/back-matter-type?post=27"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-json\/wp\/v2\/contributor?post=27"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/universityphysicssandbox\/wp-json\/wp\/v2\/license?post=27"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}