Saltar al contenido principal
LibreTexts Español

5.4.4E: Las Otras Funciones Trigonométricas (Ejercicios)

  1. Última actualización
  2. Guardar como PDF
  • Page ID
    116597

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    Sección 5.4 Ejercicio

    1. Si\(\theta =\dfrac{\pi \; }{4}\), encuentre valores exactos para\(\sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)\;\).
    2. Si\(\theta =\dfrac{7\pi \; }{4}\), encuentre valores exactos para\(\sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)\;\).
    3. Si\(\theta =\dfrac{5\pi \; }{6}\), encuentre valores exactos para\(\sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)\;\).
    4. Si\(\theta =\dfrac{\pi \; }{6}\), encuentre valores exactos para\(\sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)\).
    5. Si\(\theta =\dfrac{2\pi \; }{3}\), encuentre valores exactos para\(\sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)\;\).
    6. Si\(\theta =\dfrac{4\pi \; }{3}\), encuentre valores exactos para\(\sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)\).
    7. Evaluar: a.\(\sec \left(135{}^\circ \right)\) b.\(\csc \left(210{}^\circ \right)\) c.\(\tan \left(60{}^\circ \right)\) d.\(\cot \left(225{}^\circ \right)\)
    8. Evaluar: a.\(\sec \left(30{}^\circ \right)\) b.\(\csc \left(315{}^\circ \right)\) c.\(\tan \left(135{}^\circ \right)\) d.\(\cot \left(150{}^\circ \right)\)
    9. Si\(\sin \left(\theta \right)=\dfrac{3}{4}\), y\(\theta\) está en el cuadrante II, encuentra\(\cos \left(\theta \right),\; \sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)\).
    10. Si\(\sin \left(\theta \right)=\dfrac{2}{7}\), y\(\theta\) está en el cuadrante II, encuentra\(\cos \left(\theta \right),\; \sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)\).
    11. Si\(\cos \left(\theta \right)=-\dfrac{1}{3}\), y\(\theta\) está en el cuadrante III, encuentra\(\sin \left(\theta \right),\; \sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)\).
    12. Si\(\cos \left(\theta \right)=\dfrac{1}{5}\), y\(\theta\) está en el cuadrante I, encuentra\(\sin \left(\theta \right),\; \sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)\).
    13. Si\(\tan \left(\theta \right)=\dfrac{12}{5}\), y\(0\le \theta <\dfrac{\pi }{2}\), encuentra\(\sin \left(\theta \right),\; \cos \left(\theta \right),\sec \left(\theta \right),\; \csc \left(\theta \right),\; \cot \left(\theta \right)\).
    14. Si\(\tan \left(\theta \right)=4\), y\(0\le \theta <\dfrac{\pi }{2}\), encuentra\(\sin \left(\theta \right),\; \cos \left(\theta \right),\sec \left(\theta \right),\; \csc \left(\theta \right),\; \cot \left(\theta \right)\).
    15. Utilice una calculadora para encontrar seno, coseno y tangente de los siguientes valores:
      a. 0.15
      b. 4
      c. 70\(\mathrm{{}^\circ}\)
      d. 283\(\mathrm{{}^\circ}\)
    16. Utilice una calculadora para encontrar seno, coseno y tangente de los siguientes valores:
      a. 0.5
      b. 5.2
      c. 10\(\mathrm{{}^\circ}\)
      d. 195\(\mathrm{{}^\circ}\)

    Simplifica cada uno de los siguientes a una expresión que involucre una sola función trigonométrica sin fracciones.

    17. \(\csc (t)\tan \left(t\right)\)

    18. \(\cos (t)\csc \left(t\right)\)

    19. \(\dfrac{\sec \left(t\right)}{\csc \left(t\right)\; }\)

    20. \(\dfrac{\cot \left(t\right)}{\csc \left(t\right)}\)

    21. \(\dfrac{\sec \left(t\right)-\cos \left(t\right)}{\sin \left(t\right)}\)

    22. \(\dfrac{\tan \left(t\right)}{\sec \left(t\right)-\cos \left(t\right)}\)

    23. \(\dfrac{1+\cot \left(t\right)}{1+\tan \left(t\right)}\)

    24. \(\dfrac{1+\sin \left(t\right)}{1+\csc \left(t\right)}\)

    25. \(\dfrac{\sin ^{2} \left(t\right)+\cos ^{2} \left(t\right)}{\cos ^{2} \left(t\right)}\)

    26. \(\dfrac{1-\sin ^{2} \left(t\right)}{\sin ^{2} \left(t\right)}\)

    Demostrar las identidades.

    27. \(\dfrac{\sin ^{2} \left(\theta \right)}{1+\cos \left(\theta \right)} =1-\cos \left(\theta \right)\)

    28. \(\text{tan}^{2} (t) = \dfrac{1}{\text{cos}^2 (t)} - 1\)

    29. \(\text{sec}(a) - \text{cos}(a) = \text{sin}(a) \text{tan}(a)\)

    30. \(\dfrac{1 + \text{tan}^2(b)}{\text{tan}^2(b)} = \text{csc}^2(b)\)

    31. \(\dfrac{\text{csc}^2 (x) - \text{sin}^2 (x)}{\text{csc} (x) + \text{sin} (x)} = \text{cos} (x) \text{cot} (x)\)

    32. \(\dfrac{\text{sin} (\theta) - \text{cos} (\theta)}{\text{sec}(\theta) - \text{csc} (\theta)} = \text{sin} (\theta) \text{cos} (\theta)\)

    33. \(\dfrac{\text{csc}^2 (\alpha) - 1}{\text{csc}^2 (\alpha) - \text{csc} (\alpha)} = 1 + \text{sin} (\alpha)\)

    34. \(1 + \text{cot} (x) = \text{cos} (x) (\text{sec}(x) + \text{csc} (x))\)

    35. \(\dfrac{1 + \text{cos} (u)}{\text{sin} (u)} = \dfrac{\text{sin} (u)}{1 - \text{cos}(u)}\)

    36. \(2 \text{sec}^2 (t) = \dfrac{1 - \text{sin}(t)}{\text{cos}^2 (t)} + \dfrac{1}{1 -  \text{sin} (t)}\)

    37. \(\dfrac{\text{sin}^4 (\gamma) - \text{cos}^4 (\gamma)}{\text{sin} (\gamma) - \text{cos} (\gamma)} = \text{sin} (\gamma) + \text{cos} (\gamma)\)

    38. \(\dfrac{(1 + \text{cos}(A))(1 - \text{cos} (A))}{\text{sin} (A)} = \text{sin} (A)\)

    Contestar

    1. \(\text{sec} (\theta) = \sqrt{2}\),\(\text{csc} (\theta) = \sqrt{2}\),\(\text{tan} (\theta) = 1\),\(\text{cot} (\theta) = 1\)

    3. \(\text{sec} (\theta) = -\dfrac{2\sqrt{3}}{3}\),\(\text{csc} (\theta) = 2\),\(\text{tan} (\theta) = -\dfrac{\sqrt{3}}{3}\),\(\text{cot} (\theta) = -\sqrt{3}\)

    5. \(\text{sec} (\theta) = -2\),\(\text{csc} (\theta) = \dfrac{2\sqrt{3}}{3}\),\(\text{tan} (\theta) = -\sqrt{3}\),\(\text{cot} (\theta) = -\dfrac{\sqrt{3}}{3}\)

    7. a.\(\text{sec} (135^{\circ}) = -\sqrt{2}\)
    b.\(\text{csc} (210^{\circ}) = -2\)
    c.\(\text{tan} (60^{\circ}) = \sqrt{3}\)
    d.\(\text{cot} (225^{\circ}) = 1\)

    9. \(\cos(\theta) = -\dfrac{\sqrt{7}}{4}\),\(\sec (\theta) = -\dfrac{4\sqrt{7}}{7}\)\(\csc(\theta) = \dfrac{4}{3}\),\(\tan(\theta) = -\dfrac{3\sqrt{7}}{7}\),\(\cot(\theta) = -\dfrac{\sqrt{7}}{3}\)

    11. \(\sin(\theta) = -\dfrac{2\sqrt{2}}{3}\),\(\csc(\theta) = -\dfrac{3\sqrt{2}}{3}\)\(\sec(\theta) = -3\),\(\tan(\theta) = 2\sqrt{2}\),\(\cot(\theta) = \dfrac{\sqrt{2}}{4}\)

    13. \(\sin(\theta) = \dfrac{12}{13}\),\(\cos(\theta) = \dfrac{5}{13}\)\(\sec(\theta) = \dfrac{13}{5}\),\(\csc(\theta) = \dfrac{13}{12}\),\(\cot(\theta) = \dfrac{5}{12}\)

    15. a. sin (0.15) = 0.1494 cos (0.15) = 0.9888 bronceado (0.15) = 0.1511
    b. sin (4) = -0.7568 cos (4) = -0.6536 bronceado (4) = 1.1578
    c. sin (\(70^{\circ}\)) = 0.9397 cos (\(70^{\circ}\)) = 0.3420 tan (\(70^{\circ}\)) = 2.7475
    d. sin (\(283^{\circ}\)) = -0.9744 cos (\(283^{\circ}\)) = 0.2250 tan (\(283^{\circ}\)) = -4.3315

    17. seg (\(t\))

    19. bronceado (\(t\))

    21. bronceado (\(t\))

    23. cuna (\(t\))

    25. \((\sec(t))^2\)

    This page titled 5.4.4E: Las Otras Funciones Trigonométricas (Ejercicios) is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by David Lippman & Melonie Rasmussen (The OpenTextBookStore) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.