Saltar al contenido principal
LibreTexts Español

A.1: Trigonometría

  • Page ID
    118873
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    \( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

    ( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\id}{\mathrm{id}}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\kernel}{\mathrm{null}\,}\)

    \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\)

    \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\)

    \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    \( \newcommand{\vectorA}[1]{\vec{#1}}      % arrow\)

    \( \newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow\)

    \( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vectorC}[1]{\textbf{#1}} \)

    \( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

    \( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

    \( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    A.1.1 Trigonometría — Gráficas

    image-847.svg
    \[ \sin \theta \nonumber \]
    image-848.svg
    \[ \cos \theta \nonumber \]
    image-849.svg
    \[ \tan \theta \nonumber \]
     

    A.1.2 Trigonometría — Triángulos Especiales

    special_triangles.svg

    Del par anterior de triángulos especiales tenemos

    \[\begin{align*} \sin \frac{\pi}{4} &= \frac{1}{\sqrt{2}} & \sin \frac{\pi}{6} &= \frac{1}{2} & \sin \frac{\pi}{3} &= \frac{\sqrt{3}}{2}\\ \cos \frac{\pi}{4} &= \frac{1}{\sqrt{2}} & \cos \frac{\pi}{6} &= \frac{\sqrt{3}}{2} & \cos \frac{\pi}{3} &= \frac{1}{2}\\ \tan \frac{\pi}{4} &= 1 & \tan \frac{\pi}{6} &= \frac{1}{\sqrt{3}} & \tan \frac{\pi}{3} &= \sqrt{3} \end{align*}\]

    A.1.3 Trigonometría — Identidades simples

    • Periodicidad

      \[\begin{align*} \sin(\theta+2\pi) &= \sin(\theta) & \cos(\theta+2\pi) &= \cos(\theta) \end{align*}\]

    • Reflexión

      \[\begin{align*} \sin(-\theta)&=-\sin(\theta) & \cos(-\theta) &=\cos(\theta) \end{align*}\]

    • Reflexión alrededor\(\pi/4\)

      \[\begin{align*} \sin\left(\tfrac{\pi}{2}-\theta\right)&=\cos\theta & \cos\left(\tfrac{\pi}{2}-\theta\right)&=\sin\theta \end{align*}\]

    • Reflexión alrededor\(\pi/2\)

      \[\begin{align*} \sin\left(\pi-\theta\right)&=\sin\theta & \cos\left(\pi-\theta\right)&=-\cos\theta \end{align*}\]

    • Rotación por\(\pi\)

      \[\begin{align*} \sin\left(\theta+\pi\right)&=-\sin\theta & \cos\left(\theta+\pi\right)&=-\cos\theta \end{align*}\]

    • Pitágoras

      \[\begin{align*} \sin^2\theta + \cos^2 \theta &=1\\ \tan^2\theta + 1 &= \sec^2\theta\\ 1 + \cot^2 \theta &=\csc^2\theta \end{align*}\]

    • \(\sin\)y bloques\(\cos\) de construcción

      \[\begin{gather*} \tan\theta=\frac{\sin\theta}{\cos\theta}\quad \csc\theta=\frac{1}{\sin\theta}\quad \sec\theta=\frac{1}{\cos\theta}\quad \cot\theta=\frac{\cos\theta}{\sin\theta}=\frac{1}{\tan\theta} \end{gather*}\]

    A.1.4 Trigonometría — Sumar y restar ángulos

    • Sine

      \[\begin{align*} \sin(\alpha \pm \beta) &= \sin(\alpha )\cos(\beta) \pm \cos(\alpha )\sin(\beta) \end{align*}\]

    • Coseno

      \[\begin{align*} \cos(\alpha \pm \beta) &= \cos(\alpha )\cos(\beta) \mp \sin(\alpha )\sin(\beta) \end{align*}\]

    • Tangente

      \[\begin{align*} \tan(\alpha +\beta)&=\frac{\tan\alpha +\tan\beta}{1-\tan\alpha \tan\beta}\\ \tan(\alpha -\beta)&=\frac{\tan\alpha -\tan\beta}{1+\tan\alpha \tan\beta} \end{align*}\]

    • Doble ángulo

      \[\begin{align*} \sin(2\theta) &= 2\sin(\theta)\cos(\theta)\\ \cos(2\theta) &= \cos^2(\theta) - \sin^2(\theta)\\ &= 2\cos^2(\theta) - 1\\ &= 1 - 2\sin^2(\theta)\\ \tan(2\theta) &= \frac{2\tan(\theta)}{1-\tan^2\theta}\\ \cos^2\theta&=\frac{1+\cos(2\theta)}{2}\\ \sin^2\theta&=\frac{1-\cos(2\theta)}{2}\\ \tan^2\theta&=\frac{1-\cos(2\theta)}{1+\cos(2\theta)} \end{align*}\]

    • Productos a sumas

      \[\begin{align*} \sin(\alpha )\cos(\beta)&= \frac{\sin(\alpha +\beta) + \sin(\alpha -\beta)}{2}\\ \sin(\alpha )\sin(\beta)&= \frac{\cos(\alpha -\beta) - \cos(\alpha +\beta)}{2}\\ \cos(\alpha )\cos(\beta)&= \frac{\cos(\alpha -\beta) + \cos(\alpha +\beta)}{2} \end{align*}\]

    • Sumas a productos

      \[\begin{align*} \sin\alpha +\sin\beta &= 2 \sin\frac{\alpha +\beta}{2}\cos\frac{\alpha -\beta}{2}\\ \sin\alpha -\sin\beta &= 2 \cos\frac{\alpha +\beta}{2}\sin\frac{\alpha -\beta}{2}\\ \cos\alpha +\cos\beta &= 2 \cos\frac{\alpha +\beta}{2}\cos\frac{\alpha -\beta}{2}\\ \cos\alpha -\cos\beta &= -2 \sin\frac{\alpha +\beta}{2}\sin\frac{\alpha -\beta}{2} \end{align*}\]

    A.1.5 Funciones trigonométricas inversas

    image-851.svg

    \[ \arcsin x \nonumber \]

    Dominio:\(-1 \leq x \leq 1\)

    Rango:\(-\frac{\pi}{2} \leq \arcsin x \leq \frac{\pi}{2}\)

    image-852.svg

    \[ \arccos x \nonumber \]

    Dominio:\(-1 \leq x \leq 1\)

    Rango:\(0 \leq \arccos x \leq \pi\)

    image-853.svg

    \(\arctan x \nonumber\)

    Dominio: todos los números reales

    Rango:\(-\frac{\pi}{2} \lt \arctan x \lt \frac{\pi}{2}\)

    Como estas funciones son inversas unas de otras tenemos

    \[\begin{align*} \arcsin(\sin \theta) &= \theta & -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}\\ \arccos(\cos \theta) &= \theta & 0 \leq \theta \leq \pi\\ \arctan(\tan \theta) &= \theta & -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2} \end{align*}\]

    y también

    \[\begin{align*} \sin(\arcsin x) &= x & -1 \leq x \leq 1\\ \cos(\arccos x) &= x & -1 \leq x \leq 1\\ \tan(\arctan x) &= x & \text{any real } x \end{align*}\]

    image-854.svg

    \[ \textrm{arccsc} x \nonumber \]

    Dominio:\(|x|\ge 1\)

    Rango:\(-\frac{\pi}{2} \leq \textrm{arccsc} x \leq \frac{\pi}{2}\)

    \[ \textrm{arccsc} x \ne 0 \nonumber \]

    image-855.svg

    \[ \textrm{arcsec} x \nonumber \]

    Dominio:\(|x|\ge 1\)

    Rango:\(0 \leq \textrm{arcsec} x \leq \pi\)

    \[ \textrm{arcsec} x \ne \frac{\pi}{2} \nonumber \]

    image-856.svg

    \[ \textrm{arccot} x \nonumber \]

    Dominio: todos los números reales

    Rango:\(0 \lt \textrm{arccot} x \lt \pi\)

     

    Nuevamente

    \[\begin{align*} \textrm{arccsc}(\csc \theta) &= \theta & -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2},\ \theta\ne 0\\ \textrm{arcsec}(\sec \theta) &= \theta & 0 \leq \theta \leq \pi,\ \theta\ne \frac{\pi}{2}\\ \textrm{arccot}(\cot \theta) &= \theta & 0 \lt \theta \lt \pi \end{align*}\]

    y

    \[\begin{align*} \csc(\textrm{arccsc} x) &= x & |x|\ge 1\\ \sec(\textrm{arcsec} x) &= x & |x|\ge 1\\ \cot(\textrm{arccot} x) &= x & \text{any real } x \end{align*}\]


    This page titled A.1: Trigonometría is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Joel Feldman, Andrew Rechnitzer and Elyse Yeager via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.