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).