$$\require{cancel}$$

# Appendix E: Mathematical Formulas

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).