5.4.4E: Las Otras Funciones Trigonométricas (Ejercicios)
- Page ID
- 116597
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Sección 5.4 Ejercicio
- Si\(\theta =\dfrac{\pi \; }{4}\), encuentre valores exactos para\(\sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)\;\).
- Si\(\theta =\dfrac{7\pi \; }{4}\), encuentre valores exactos para\(\sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)\;\).
- Si\(\theta =\dfrac{5\pi \; }{6}\), encuentre valores exactos para\(\sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)\;\).
- Si\(\theta =\dfrac{\pi \; }{6}\), encuentre valores exactos para\(\sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)\).
- Si\(\theta =\dfrac{2\pi \; }{3}\), encuentre valores exactos para\(\sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)\;\).
- Si\(\theta =\dfrac{4\pi \; }{3}\), encuentre valores exactos para\(\sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)\).
- Evaluar: a.\(\sec \left(135{}^\circ \right)\) b.\(\csc \left(210{}^\circ \right)\) c.\(\tan \left(60{}^\circ \right)\) d.\(\cot \left(225{}^\circ \right)\)
- Evaluar: a.\(\sec \left(30{}^\circ \right)\) b.\(\csc \left(315{}^\circ \right)\) c.\(\tan \left(135{}^\circ \right)\) d.\(\cot \left(150{}^\circ \right)\)
- Si\(\sin \left(\theta \right)=\dfrac{3}{4}\), y\(\theta\) está en el cuadrante II, encuentra\(\cos \left(\theta \right),\; \sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)\).
- Si\(\sin \left(\theta \right)=\dfrac{2}{7}\), y\(\theta\) está en el cuadrante II, encuentra\(\cos \left(\theta \right),\; \sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)\).
- Si\(\cos \left(\theta \right)=-\dfrac{1}{3}\), y\(\theta\) está en el cuadrante III, encuentra\(\sin \left(\theta \right),\; \sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)\).
- Si\(\cos \left(\theta \right)=\dfrac{1}{5}\), y\(\theta\) está en el cuadrante I, encuentra\(\sin \left(\theta \right),\; \sec \left(\theta \right),\csc \left(\theta \right),\; \tan \left(\theta \right),\; \cot \left(\theta \right)\).
- Si\(\tan \left(\theta \right)=\dfrac{12}{5}\), y\(0\le \theta <\dfrac{\pi }{2}\), encuentra\(\sin \left(\theta \right),\; \cos \left(\theta \right),\sec \left(\theta \right),\; \csc \left(\theta \right),\; \cot \left(\theta \right)\).
- Si\(\tan \left(\theta \right)=4\), y\(0\le \theta <\dfrac{\pi }{2}\), encuentra\(\sin \left(\theta \right),\; \cos \left(\theta \right),\sec \left(\theta \right),\; \csc \left(\theta \right),\; \cot \left(\theta \right)\).
- Utilice una calculadora para encontrar seno, coseno y tangente de los siguientes valores:
a. 0.15
b. 4
c. 70\(\mathrm{{}^\circ}\)
d. 283\(\mathrm{{}^\circ}\) - 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 (\(283^{\circ}\)) = -4.331517. seg (\(t\))
19. bronceado (\(t\))
21. bronceado (\(t\))
23. cuna (\(t\))
25. \((\sec(t))^2\)