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Appendix E: Mathematical Formulas

  • Page ID
    7936
  • Quadratic formula

    If ax2 + bx + c = 0, then x = \(\frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\)

    Table E1 - Geometry

    Triangle of base b and height h Area = \(\frac{1}{2}\)bh  
    Circle of radius r Circumference = 2\(\pi\)r Area = \(\pi\)r2
    Sphere of radius r Surface Area = 2\(\pi\)r2 Volume = \(\frac{4}{3} \pi\)r3
    Cylinder of radius r and height h Area of curved surface = 2\(\pi\)rh Volume = \(\pi\)r2h

    Trigonometry

    Trigonometric Identities

    1. sin \(\theta\) = \(\frac{1}{\csc \theta}\) 
    2. cos \(\theta\) = \(\frac{1}{\sec \theta}\)
    3. tan \(\theta\) = \(\frac{1}{\cot \theta}\)
    4. sin(90° − \(\theta\)) = cos \(\theta\)
    5. cos(90° − \(\theta\)) = sin \(\theta\)
    6. tan(90° − \(\theta\)) = cot \(\theta\)
    7. sin2 \(\theta\) + cos2 \(\theta\) = 1
    8. sec2 \(\theta\) − tan2 \(\theta\) = 1
    9. tan \(\theta\) = \(\frac{\sin \theta}{\cos \theta}\)
    10. sin(\(\alpha \pm \beta\)) = sin \(\alpha\) cos \(\beta\) ± cos \(\alpha\) sin \(\beta\) 
    11. cos(\(\alpha \pm \beta\)) = cos \(\alpha\) cos \(\beta\) ∓ sin \(\alpha\) sin \(\beta\)
    12. tan(\(\alpha \pm \beta\)) = \(\frac{\tan \alpha \pm \tan \beta}{1 \mp \tan \alpha \tan \beta}\) 
    13. sin 2\(\theta\) = 2sin \(\theta\)cos \(\theta\)
    14. cos 2\(\theta\) = cos2 \(\theta\) − sin2 \(\theta\) = 2 cos2 \(\theta\) − 1 = 1 − 2 sin2 \(\theta\)
    15. sin \(\alpha\) + sin \(\beta\) = 2 sin\(\frac{1}{2}\)(\(\alpha\) + \(\beta\))cos\(\frac{1}{2}\)(\(\alpha\) − \(\beta\))
    16. cos \(\alpha\) + cos \(\beta\) = 2 cos\(\frac{1}{2}\)(\(\alpha\) + \(\beta\))cos\(\frac{1}{2}\)(\(\alpha\) − \(\beta\))

    Triangles

    1. Law of sines: \(\frac{a}{\sin \alpha}\) = \(\frac{b}{\sin \beta}\) = \(\frac{c}{\sin \gamma}\)
    2. Law of cosines: c2 = a2 + b2 − 2ab cos \(\gamma\)

    1. Pythagorean theorem: a2 + b2 = c2

    Series expansions

    1. Binomial theorem: (a + b)n = an + nan-1b + \(\frac{n(n-1)a^{n-2} b^{2}}{2!}\) + \(\frac{n(n-1)(n-2)a^{n-3} b^{3}}{3!}\) + \(\cdots\)
    2. (1 ± x)n = 1 ± \(\frac{nx}{1!} + \frac{n(n-1)x^{2}}{2!} \pm \cdots\) (x2 < 1)
    3. (1 ± x)-n = 1 ∓ \(\frac{nx}{1!} + \frac{n(n+1)x^{2}}{2!} \mp \cdots\) (x2 < 1)
    4. sin x = x - \(\frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \cdots\)
    5. cos x = 1 - \(\frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \cdots\)
    6. tan x = x + \(\frac{x^{3}}{3} + \frac{2x^{5}}{15} + \cdots\)
    7. ex = 1 + x + \(\frac{x^{2}}{2!} + \cdots\)
    8. ln(1 + x) = x − \(\frac{1}{2}x^{2} + \frac{1}{3}x^{3} − \cdots\) (|x| < 1)

    Derivatives

    1. \(\frac{d}{dx}\)[a f(x)] = a \(\frac{d}{dx}\)f(x)
    2. \(\frac{d}{dx}\)[f(x) + g(x)] = \(\frac{d}{dx}\)f(x) + \(\frac{d}{dx}\)g(x)
    3. \(\frac{d}{dx}\)[f(x)g(x)] = f(x) \(\frac{d}{dx}\)g(x) + g(x) \(\frac{d}{dx}\)f(x)
    4. \(\frac{d}{dx}\)f(u) = [\(\frac{d}{du}\)f(u)]\(\frac{du}{dx}\)
    5. \(\frac{d}{dx}\)xm = mxm − 1
    6. \(\frac{d}{dx}\)sin x = cos x
    7. \(\frac{d}{dx}\)cos x = −sin x
    8. \(\frac{d}{dx}\)tan x = sec2 x
    9. \(\frac{d}{dx}\)cot x = −csc2 x
    10. \(\frac{d}{dx}\)sec x = tan x sec x
    11. \(\frac{d}{dx}\)csc x = −cot x csc x
    12. \(\frac{d}{dx}\)ex = ex
    13. \(\frac{d}{dx}\)ln x = \(\frac{1}{x}\)
    14. \(\frac{d}{dx}\)sin−1 x = \(\frac{1}{1 − x^{2}}\) 
    15. \(\frac{d}{dx}\)cos−1 x = \(− \frac{1}{1 − x^{2}}\)
    16. \(\frac{d}{dx}\)tan−1 x = \(− \frac{1}{1 + x^{2}}\)

    Integrals

    1. \(\int\)a f(x)dx = a \(\int\)f(x)dx
    2. \(\int\)[f(x) + g(x)]dx = \(\int\)f(x)dx + \(\int\)g(x)dx
    3. \(\int\)xm dx = \(\frac{x^{m + 1}}{m + 1}\) for (m ≠ −1) = ln x for (m = −1)
    4. \(\int\)sin x dx = −cos x
    5. \(\int\)cos x dx = sin x
    6. \(\int\)tan x dx = ln|sec x|
    7. \(\int\)sin2 (ax) dx = \(\frac{x}{2}\) − \(\frac{\sin 2ax}{4a}\)
    8. \(\int\)cos2 (ax) dx = \(\frac{x}{2}\) + \(\frac{\sin 2ax}{4a}\)
    9. \(\int\)sin (ax) cos (ax) dx = \(− \frac{\cos 2ax}{4a}\)
    10. \(\int\)eax dx = \(\frac{1}{a}\)eax
    11. \(\int\)xeax dx = \(\frac{e^{ax}}{a^{2}}\)(ax − 1)
    12. \(\int\)ln ax dx = x ln ax − x
    13. \(\int \frac{dx}{a^{2} + x^{2}}\) = \(\frac{1}{a}\)tan−1 \(\frac{x}{a}\)
    14. \(\int \frac{dx}{a^{2} − x^{2}}\) = \(\frac{1}{2a}\) ln\(\big| \frac{x + a}{x − a} \big|\)
    15. \(\int \frac{dx}{\sqrt{a^{2} + x^{2}}}\) = sinh−1 \(\frac{x}{a}\) 
    16. \(\int \frac{dx}{\sqrt{a^{2} - x^{2}}}\) = sin−1 \(\frac{x}{a}\) 
    17. \(\int \sqrt{a^{2} + x^{2}}\) dx = \(\frac{x}{2} \sqrt{a^{2} + x^{2}} + \frac{a^{2}}{2} \sinh^{−1} \frac{x}{a}\)
    18. \(\int \sqrt{a^{2} - x^{2}}\) dx = \(\frac{x}{2} \sqrt{a^{2} - x^{2}} + \frac{a^{2}}{2} \sin^{−1} \frac{x}{a}\)

    Contributors

    • Samuel J. Ling (Truman State University), Jeff Sanny (Loyola Marymount University), and Bill Moebs with many contributing authors. This work is licensed by OpenStax University Physics under a Creative Commons Attribution License (by 4.0).