Appendix E: Mathematical Formulas

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 = \(\pi\)r^2
Sphere of radius r	Surface Area = 2\(\pi\)r^2	Volume = \(\frac{4}{3} \pi\)r^3
Cylinder of radius r and height h	Area of curved surface = 2\(\pi\)rh	Volume = \(\pi\)r^2h

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 \(\gamma\)

3. Pythagorean theorem: a² + b² = c²

Series expansions

1. Binomial theorem: (a + b)ⁿ = aⁿ + na^n-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. e^x = 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. \(\frac{d}{dx}\)x^m = mx^{m − 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. \(\frac{d}{dx}\)e^x = e^x
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. \(\int\)x^m 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. \(\int\)e^ax dx = \(\frac{1}{a}\)e^ax
11. \(\int\)xe^ax 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 and Attributions

• 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).