Respuestas - Capítulo 5

Ejercicios 1 al 8.

  1. $$$$
    2. $\text{Complementario:\;}90^{\circ}-30^{\circ}=60^{\circ}\\[0.1cm] \text{Suplementario:}\;180^{\circ}-30^{\circ}=150^{\circ}\\[0.1cm] \text{Coterminal:}\;360^{\circ}+30^{\circ}=390^{\circ}\\[0.3cm] 30^{\circ}=30^{\circ}\cdot\cfrac{\pi}{180^{\circ}}=\cfrac{\pi}{6}$
  2. $$$$
    3. $\text{Complementario:\;}90^{\circ}-45^{\circ}=45^{\circ}\\[0.1cm] \text{Suplementario:}\;180^{\circ}-45^{\circ}=135^{\circ}\\[0.1cm] \text{Coterminal:}\;360^{\circ}+45^{\circ}=405^{\circ}\\[0.3cm] 45^{\circ}=45^{\circ}\cdot\cfrac{\pi}{180^{\circ}}=\cfrac{\pi}{4}$
  3. $$$$
    4. $\text{Complementario:\;}90^{\circ}-60^{\circ}=30^{\circ}\\[0.1cm] \text{Suplementario:}\;180^{\circ}-60^{\circ}=120^{\circ}\\[0.1cm] \text{Coterminal:}\;360^{\circ}+60^{\circ}=420^{\circ}\\[0.3cm] 60^{\circ}=60^{\circ}\cdot\cfrac{\pi}{180^{\circ}}=\cfrac{\pi}{3}$
  4. $$$$
    5. $\text{Complementario:No se calcula el ángulo mide más de} \;90^{\circ}\\[0.1cm] \text{Suplementario:}\;180^{\circ}-120^{\circ}=30^{\circ}\\[0.1cm] \text{Coterminal:}\;360^{\circ}+120^{\circ}=480^{\circ}\\[0.3cm] 120^{\circ}=120^{\circ}\cdot\cfrac{\pi}{180^{\circ}}=\cfrac{2\pi}{3}$
  5. $$$$
    6. $\text{Complementario:\;}\cfrac{\pi}{2}-\cfrac{\pi}{9}=\cfrac{7\pi}{18}\\[0.1cm] \text{Suplementario:}\;\pi-\cfrac{\pi}{9}=\cfrac{8\pi}{9}\\[0.1cm] \text{Coterminal:}\;2\pi+\cfrac{\pi}{9}=\cfrac{19\pi}{9}\\[0.3cm] \cfrac{\pi}{9}=\cfrac{\pi}{9}\cdot\cfrac{180^{\circ}}{\pi}=20^{\circ}$
  1. $$$$
    2. $\text{Complementario:\;}\cfrac{\pi}{2}-\cfrac{37\pi}{180}=\cfrac{53\pi}{180}\\[0.3cm] \text{Suplementario:}\;\pi-\cfrac{37\pi}{180}=\cfrac{143\pi}{180}\\[0.3cm] \text{Coterminal:}\;2\pi+\cfrac{37\pi}{180}=\cfrac{397\pi}{180}\\[0.3cm] \cfrac{37\pi}{180}=\cfrac{37\pi}{180}\cdot\cfrac{180^{\circ}}{\pi}=37^{\circ}$


  2. $$$$
    3. $\text{Complementario:\;}\cfrac{\pi}{2}-\cfrac{5\pi}{18}=\cfrac{2\pi}{9}\\[0.3cm] \text{Suplementario:}\;\pi-\cfrac{5\pi}{18}=\cfrac{13\pi}{18}\\[0.3cm] \text{Coterminal:}\;2\pi+\cfrac{5\pi}{18}=\cfrac{41\pi}{18}\\[0.3cm] \cfrac{5\pi}{18}=\cfrac{5\pi}{18}\cdot\cfrac{180^{\circ}}{\pi}=50^{\circ}$


  3. $$$$
    4. $\text{Complementario:\;}\cfrac{\pi}{2}-\cfrac{53\pi}{180}=\cfrac{37\pi}{180}\\[0.3cm] \text{Suplementario:}\;\pi-\cfrac{53\pi}{180}=\cfrac{127\pi}{180}\\[0.3cm] \text{Coterminal:}\;2\pi+\cfrac{53\pi}{180}=\cfrac{413\pi}{180}\\[0.3cm] \cfrac{53\pi}{180}=\cfrac{53\pi}{180}\cdot\cfrac{180^{\circ}}{\pi}=53^{\circ}$

Ejercicios 9 al 16.

  1. Pertenece
  2. No pertenece
  1. $y=\cfrac{3}{5}$
  2. $y=\cfrac{24}{25}$
  1. $H=2\wedge \alpha=30^{\circ}$
  2. $CA=\sqrt{3}\wedge \alpha=30^{\circ}$
  1. $CO=1\wedge \alpha=45^{\circ}$
  2. $H=5\wedge \alpha=37^{\circ}$

Ejercicios 17 al 24.

  1. $\alpha=30.96^{\circ} , \beta=45^{\circ},CE=5.83\; \text{pulgadas} \wedge ED=7.07\;\text{pulgadas}$.
  2. $\alpha=\delta=38.66^{\circ} , \beta=\theta=\gamma=\rho=51.34^{\circ},\sigma=77.32^{\circ}\;y\;CE=ED=6.4\; \text{pulgadas}$.
  3. $9\; \text{metros de altura y una distancia de}\;9\sqrt{3}$.
  4. $$$$
  1. $\text{a.}\; 0.0103\\[0.1cm]\text{b.}\; 0.0155$
  2. Altura $(h)=100\sqrt{3} m.\\[0.1cm]\text{ Distancia} (s)=300 m.$
  3. Distancia = 7464.10 m
  4. Altura = 42.45 m

Ejercicios 25 al 32.

  1. $14.9m., 17.3m.$
  2. $10.3cm., 13.5cm.$
  3. $165.2m.$
  4. Son posibles todas comunicaciones, excepto entre Bernardo y Carlos.
  1. Juanita Martínez esta en lo correcto.
  2. $47.7\;Km$
  3. $\text{a.\;6.37 millas náuticas}\\[0.1cm] \text{b.\;}16.80^{\circ}$
  4. $AC \thickapprox\sqrt{8468}\thickapprox 92$ pies.

Ejercicios 33 al 40.

  1. $\color{black}\begin{aligned} Sen^2\;{\theta}\left(1+Cot^2\;{\theta}\right)& \equiv 1\\ Sen^2{\theta}\left(Csc^2{\theta}\right) &\equiv \\ {\color{Red}\bcancel {{\color{black}Sen^2{\theta}}}}\left(\cfrac{1}{{\color{Red}\bcancel {{\color{black}Sen^2{\theta}}}}}\right) &\equiv \\ 1 &\equiv 1 \end{aligned}$
  2. $\color{black}\begin{aligned} Tan{\theta}+Cot{\theta}& \equiv Sec\;{\theta}\cdot Csc\;{\theta}\\[0.1cm] \cfrac{Sen\;{\theta}}{Cos\;{\theta}}+\cfrac{Cos\;{\theta}}{Sen\;{\theta}} &\equiv \\[0.3cm] \cfrac{Sen^2\;\theta + Cos^2\;\theta}{Sen\;\theta \cdot Cos\;\theta} &\equiv \\[0.3cm] \cfrac{1}{Sen\;\theta \cdot Cos\;\theta} &\equiv \\[0.3cm] Sec\;{\theta}\cdot Csc\;{\theta} &\equiv Sec\;{\theta}\cdot Csc\;{\theta} \end{aligned}$
  3. $\color{black}\begin{aligned} Tan\;{\theta}+\cfrac{Cos{\theta}}{1+Sen{\theta}}& \equiv Sec\;{\theta}\\[0.1cm] \cfrac{Sen\;{\theta}}{Cos\;{\theta}}+\cfrac{Cos\;{\theta}}{1+Sen\;{\theta}} &\equiv \\[0.3cm] \cfrac{Sen\;\theta+Sen^2\;\theta + Cos^2\;\theta}{Cos\;\theta \cdot \left(1+Sen\;\theta\right)} &\equiv \\[0.3cm] \cfrac{{\color{Red}\bcancel {{\color{black}Sen\;\theta+1}}}}{Cos\;\theta \cdot \left({\color{Red}\bcancel {{\color{black}1+Sen\;\theta}}}\right)} &\equiv \\[0.3cm] Sec\;{\theta}&\equiv Sec\;{\theta} \end{aligned}$
  4. $\color{black}\begin{aligned} \cfrac{1+Sen\;{\alpha}}{1-Sen\;{\alpha}}-\cfrac{1-Sen\;{\alpha}}{1+Sen\;{\alpha}}& \equiv 4Tan\;\alpha\cdot Sec\;{\alpha}\\[0.1cm] \cfrac{\left(1+Sen\;{\alpha}\right)^2-\left(1-Sen\;{\alpha}\right)^2}{\left(1-Sen\;{\alpha}\right)\cdot \left(1+Sen\;{\alpha}\right)} &\equiv \\[0.3cm] \cfrac{{\color{Red}\bcancel {{\color{black}1}}}+2Sen\;{\alpha}+{\color{Red}\bcancel {{\color{black}Sen^2\;{\alpha}}}}-{\color{Red}\bcancel {{\color{black}1}}}+2Sen\;{\alpha}-{\color{Red}\bcancel {{\color{black}Sen^2\;{\alpha}}}}}{1-Sen^2\;{\alpha}} &\equiv \\[0.3cm] \cfrac{4Sen\;{\alpha}}{Cos^2\;{\alpha}} &\equiv \\[0.3cm] \cfrac{4Sen\;{\alpha}}{Cos\;\alpha}\cdot\cfrac{1}{Cos\;{\alpha}} &\equiv \\[0.3cm] 4Tan\;{\alpha}\cdot Sec\;\alpha &\equiv 4Tan\;{\alpha}\cdot Sec\;\alpha \end{aligned}$
  1. $\color{black}\begin{aligned} Sen\;{\theta}\left(Csc\;\theta-Sen{\theta}\right)& \equiv Cos^2\;{\theta}\\[0.1cm] Sen\;{\theta}\left(\cfrac{1}{Sen\;\theta}-Sen{\theta}\right) &\equiv \\[0.3cm] {\color{Red}\bcancel {{\color{black}Sen\;{\theta}}}}\left(\cfrac{1-Sen^2\;\theta}{{\color{Red}\bcancel {{\color{black}Sen\;\theta}}}}\right) &\equiv \\[0.3cm] Cos^2\;{\theta}&\equiv Cos^2\;{\theta} \end{aligned}$
  2. $\color{black}\begin{aligned} \left(1-Cos^2\;{\theta}\right)\left(1+Cot^2{\theta}\right)& \equiv 1\\[0.1cm] \left(Sen^2\;{\theta}\right)\left(Csc^2{\theta}\right) &\equiv \\[0.3cm] {\color{Red}\bcancel {{\color{black}Sen^2\;{\theta}}}}\left(\cfrac{1}{{\color{Red}\bcancel {{\color{black}Sen^2\;\theta}}}}\right) &\equiv \\[0.3cm] 1&\equiv 1 \end{aligned}$
  3. $\color{black}\begin{aligned} Cos\;{\beta}\left(Tan\;\beta+Cot{\beta}\right)& \equiv Csc\;{\beta}\\[0.1cm] Cos\;{\beta}\left(\cfrac{Sen\;\beta}{Cos\;\beta}+\cfrac{Cos\;\beta}{Sen\;\beta}\right) &\equiv \\[0.3cm] {\color{Red}\bcancel {{\color{black}Cos\;{\beta}}}}\left(\cfrac{Sen^2\;\beta + Cos^2\;\beta}{{\color{Red}\bcancel {{\color{black}Cos\;\beta}}}\cdot Sen\;\beta}\right) &\equiv \\[0.3cm] \cfrac{1}{Sen\;\beta} &\equiv \\[0.3cm] Csc\;{\beta}&\equiv Csc\;{\beta} \end{aligned}$
  4. $\color{black}\begin{aligned} Sec{\theta}-Tan{\theta}& \equiv \cfrac{1-Sen\;{\theta}}{Cos\;{\theta}} \\[0.1cm] \cfrac{1}{Cos\;{\theta}}-\cfrac{Sen\;{\theta}}{Cos\;{\theta}} &\equiv \\[0.3cm] \cfrac{1-Sen\;\theta}{Cos\;{\theta}} &\equiv \cfrac{1-Sen\;\theta}{Cos\;{\theta}} \end{aligned}$

Ejercicios 41 al 52.

  1. $\hspace{0.1cm}\cfrac{7\pi}{6}+NV\left(2\pi\right)$
  2. $\hspace{0.1cm}\cfrac{\pi}{3}+NV\left(2\pi\right), \cfrac{5\pi}{3}+NV\left(2\pi\right),\\\pi+NV\left(2\pi\right)$
  3. $\hspace{0.1cm}\cfrac{\pi}{12}+NV\left(\pi\right),\cfrac{5\pi}{12}+NV\left(\pi\right)$
  4. $\hspace{0.1cm}\cfrac{-\pi}{12}+NV\left(2\pi\right),\cfrac{7\pi}{12}+NV\left(2\pi\right)$
  1. $\hspace{0.1cm}\cfrac{\pi}{2}+NV\left(\pi\right),\cfrac{2\pi}{3}+NV\left(\pi\right),\\\cfrac{5\pi}{6}+NV\left(\pi\right)$
  2. $\hspace{0.1cm}NV\left(360^{\circ}\right)$
  3. $\hspace{0.1cm}120^{\circ}+NV\left(360^{\circ}\right), 240^{\circ}+NV\left(360^{\circ}\right)$
  4. $\hspace{0.1cm}NV\left(\pi\right),\cfrac{2\pi}{3}+NV\left(2\pi\right),\\\cfrac{4\pi}{3}+NV\left(2\pi\right)$
  1. $\hspace{0.1cm}\cfrac{4\pi}{3}+NV\left(2\pi\right),\cfrac{5\pi}{3}+NV\left(2\pi\right)$
  2. $\hspace{0.1cm}\cfrac{\pi}{4}+NV\left(\pi\right),\cfrac{7\pi}{12}+NV\left(\pi\right)$
  3. $\hspace{0.1cm}NV\left(\pi\right)$
  4. $\hspace{0.1cm}NV\left(2\pi\right),\cfrac{\pi}{2}+NV\left(2\pi\right)$