C Repaso de Precálculo

Leyes de los exponentes

Los términos $\begin{array}{ccccccccccccc}\hfill x^m{x}^n& =\hfill & x^{m+n}\hfill & & & \hfill \frac{x^m}{x^n}& =\hfill & x^{m-n}\hfill & & & \hfill {\left(x^m\right)}^n& =\hfill & x^{mn}\hfill \\ \hfill x^{\text{−}n}& =\hfill & \frac{1}{x^n}\hfill & & & \hfill {\left(xy\right)}^n& =\hfill & x^n{y}^n\hfill & & & \hfill {\left(\frac{x}{y}\right)}^n& =\hfill & \frac{x^n}{y^n}\hfill \\ \hfill x^{1\text{/}n}& =\hfill & \sqrt[n]{x}\hfill & & & \hfill \sqrt[n]{xy}& =\hfill & \sqrt[n]{x}\sqrt[n]{y}\hfill & & & \hfill \sqrt[n]{\frac{x}{y}}& =\hfill & \frac{\sqrt[n]{x}}{\sqrt[n]{y}}\hfill \\ \hfill x^{m\text{/}n}& =\hfill & \sqrt[n]{x^m}={\left(\sqrt[n]{x}\right)}^m\hfill & & & & & & & & & & \end{array}$

Factorizaciones especiales

Los términos $\begin{array}{ccc}\hfill x^2-y^2& =\hfill & \left(x+y\right)\left(x-y\right)\hfill \\ \hfill x^3+y^3& =\hfill & \left(x+y\right)\left(x^2-xy+y^2\right)\hfill \\ \hfill x^3-y^3& =\hfill & \left(x-y\right)\left(x^2+xy+y^2\right)\hfill \end{array}$

Fórmula cuadrática

Si los valores de $a{x}^2+bx+c=0,$ entonces $x=\frac{\text{−}b\pm \sqrt{b^2-4ca}}{2a}.$

Teorema del binomio

Los términos ${\left(a+b\right)}^n=a^n+\left(\begin{array}{l}n\\ 1\end{array}\right)a^{n–1}b+\left(\begin{array}{l}n\\ 2\end{array}\right)a^{n–2}{b}^2+\cdots +\left(\begin{array}{c}n\\ n–1\end{array}\right)a{b}^{n–1}+b^n,$

donde $\left(\begin{array}{l}n\\ k\end{array}\right)=\frac{n\left(n–1\right)\left(n–2\right)\cdots \left(n-k+1\right)}{k\left(k-1\right)\left(k-2\right)\cdots 3\cdot 2\cdot 1}=\frac{n!}{k!\left(n-k\right)!}$

Trigonometría de ángulo recto

Los términos $\begin{array}{cccc}\text{sen}\theta =\frac{\text{opp}}{\text{hyp}}\hfill & & & \text{csc}\theta =\frac{\text{hyp}}{\text{opp}}\hfill \\ \text{cos}\theta =\frac{\text{adj}}{\text{hyp}}\hfill & & & \text{sec}\theta =\frac{\text{hyp}}{\text{adj}}\hfill \\ \text{tan}\theta =\frac{\text{opp}}{\text{adj}}\hfill & & & \text{cot}\theta =\frac{\text{adj}}{\text{opp}}\hfill \end{array}$

Funciones trigonométricas de ángulos importantes

Identidades fundamentales

Los términos $\begin{array}{cccccccc}\hfill {\text{sen}}^2\theta +{\text{cos}}^2\theta & =\hfill & 1\hfill & & & \hfill \text{sen}\left(\text{−}\theta \right)& =\hfill & \text{−}\text{sen}\theta \hfill \\ \hfill 1+{\text{tan}}^2\theta & =\hfill & {\text{sec}}^2\theta \hfill & & & \hfill \text{cos}\left(\text{−}\theta \right)& =\hfill & \text{cos}\theta \hfill \\ \hfill 1+{\text{cot}}^2\theta & =\hfill & {\text{csc}}^2\theta \hfill & & & \hfill \text{tan}\left(\text{−}\theta \right)& =\hfill & \text{−}\text{tan}\theta \hfill \\ \hfill \text{sin}\left(\frac{\pi }{2}-\theta \right)& =\hfill & \text{cos}\theta \hfill & & & \hfill \text{sin}\left(\theta +2\pi \right)& =\hfill & \text{sen}\theta \hfill \\ \hfill \text{cos}\left(\frac{\pi }{2}-\theta \right)& =\hfill & \text{sen}\theta \hfill & & & \hfill \text{cos}\left(\theta +2\pi \right)& =\hfill & \text{cos}\theta \hfill \\ \hfill \text{tan}\left(\frac{\pi }{2}-\theta \right)& =\hfill & \text{cot}\theta \hfill & & & \hfill \text{tan}\left(\theta +\pi \right)& =\hfill & \text{tan}\theta \hfill \end{array}$

Ley de senos

Los términos $\frac{\text{sin}A}{a}=\frac{\text{sin}B}{b}=\frac{\text{sin}C}{c}$

Ley de los cosenos

Los términos $\begin{array}{ccc}\hfill a^2& =\hfill & b^2+c^2-2bc\text{cos}A\hfill \\ \hfill b^2& =\hfill & a^2+c^2-2ac\text{cos}B\hfill \\ \hfill c^2& =\hfill & a^2+b^2-2ab\text{cos}C\hfill \end{array}$

Fórmulas de suma y resta

Los términos $\begin{array}{ccc}\hfill \text{sen}\left(x+y\right)& =\hfill & \text{sen}x\text{cos}y+\text{cos}x\text{sin}y\hfill \\ \hfill \text{sin}\left(x-y\right)& =\hfill & \text{sen}x\text{cos}y-\text{cos}x\text{sin}y\hfill \\ \hfill \text{cos}\left(x+y\right)& =\hfill & \text{cos}x\text{cos}y-\text{sen}x\text{sin}y\hfill \\ \hfill \text{cos}\left(x-y\right)& =\hfill & \text{cos}x\text{cos}y+\text{sen}x\text{sin}y\hfill \\ \hfill \text{tan}\left(x+y\right)& =\hfill & \frac{\text{tan}x+\text{tan}y}{1-\text{tan}x\text{tan}y}\hfill \\ \hfill \text{tan}\left(x-y\right)& =\hfill & \frac{\text{tan}x-\text{tan}y}{1+\text{tan}x\text{tan}y}\hfill \end{array}$

Fórmulas del ángulo doble

Los términos $\begin{array}{ccc}\hfill \text{sen}2x& =\hfill & 2\text{sen}x\text{cos}x\hfill \\ \hfill \text{cos}2x& =\hfill & {\text{cos}}^{2}x-{\text{sen}}^{2}x=2{\text{cos}}^{2}x–1=1-2{\text{sen}}^{2}x\hfill \\ \hfill \text{tan}2x& =\hfill & \frac{2\text{tan}x}{1-{\text{tan}}^{2}x}\hfill \end{array}$

Fórmulas de ángulo mitad

Los términos $\begin{array}{ccc}\hfill {\text{sen}}^{2}x& =\hfill & \frac{1-\text{cos}2x}{2}\hfill \\ \hfill {\text{cos}}^{2}x& =\hfill & \frac{1+\text{cos}2x}{2}\hfill \end{array}$