Respuestas - Capítulo 5

Ejercicios 1 al 8.

  1. $$$$
    2. Complementario:  90∘−30∘=60∘Suplementario:  180∘−30∘=150∘Coterminal:  360∘+30∘=390∘30∘=30∘⋅π180∘=π6\text{Complementario:\;}90^{\circ}-30^{\circ}=60^{\circ}\\[0.1cm] \text{Suplementario:}\;180^{\circ}-30^{\circ}=150^{\circ}\\[0.1cm] \text{Coterminal:}\;360^{\circ}+30^{\circ}=390^{\circ}\\[0.3cm] 30^{\circ}=30^{\circ}\cdot\cfrac{\pi}{180^{\circ}}=\cfrac{\pi}{6}
  2. $$$$
    3. Complementario:  90∘−45∘=45∘Suplementario:  180∘−45∘=135∘Coterminal:  360∘+45∘=405∘45∘=45∘⋅π180∘=π4\text{Complementario:\;}90^{\circ}-45^{\circ}=45^{\circ}\\[0.1cm] \text{Suplementario:}\;180^{\circ}-45^{\circ}=135^{\circ}\\[0.1cm] \text{Coterminal:}\;360^{\circ}+45^{\circ}=405^{\circ}\\[0.3cm] 45^{\circ}=45^{\circ}\cdot\cfrac{\pi}{180^{\circ}}=\cfrac{\pi}{4}
  3. $$$$
    4. Complementario:  90∘−60∘=30∘Suplementario:  180∘−60∘=120∘Coterminal:  360∘+60∘=420∘60∘=60∘⋅π180∘=π3\text{Complementario:\;}90^{\circ}-60^{\circ}=30^{\circ}\\[0.1cm] \text{Suplementario:}\;180^{\circ}-60^{\circ}=120^{\circ}\\[0.1cm] \text{Coterminal:}\;360^{\circ}+60^{\circ}=420^{\circ}\\[0.3cm] 60^{\circ}=60^{\circ}\cdot\cfrac{\pi}{180^{\circ}}=\cfrac{\pi}{3}
  4. $$$$
    5. Complementario:No se calcula el aˊngulo mide maˊs de  90∘Suplementario:  180∘−120∘=30∘Coterminal:  360∘+120∘=480∘120∘=120∘⋅π180∘=2π3\text{Complementario:No se calcula el ángulo mide más de} \;90^{\circ}\\[0.1cm] \text{Suplementario:}\;180^{\circ}-120^{\circ}=30^{\circ}\\[0.1cm] \text{Coterminal:}\;360^{\circ}+120^{\circ}=480^{\circ}\\[0.3cm] 120^{\circ}=120^{\circ}\cdot\cfrac{\pi}{180^{\circ}}=\cfrac{2\pi}{3}
  5. $$$$
    6. Complementario:  π2−π9=7π18Suplementario:  π−π9=8π9Coterminal:  2π+π9=19π9π9=π9⋅180∘π=20∘\text{Complementario:\;}\cfrac{\pi}{2}-\cfrac{\pi}{9}=\cfrac{7\pi}{18}\\[0.1cm] \text{Suplementario:}\;\pi-\cfrac{\pi}{9}=\cfrac{8\pi}{9}\\[0.1cm] \text{Coterminal:}\;2\pi+\cfrac{\pi}{9}=\cfrac{19\pi}{9}\\[0.3cm] \cfrac{\pi}{9}=\cfrac{\pi}{9}\cdot\cfrac{180^{\circ}}{\pi}=20^{\circ}
  1. $$$$
    2. Complementario:  π2−37π180=53π180Suplementario:  π−37π180=143π180Coterminal:  2π+37π180=397π18037π180=37π180⋅180∘π=37∘\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. Complementario:  π2−5π18=2π9Suplementario:  π−5π18=13π18Coterminal:  2π+5π18=41π185π18=5π18⋅180∘π=50∘\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. Complementario:  π2−53π180=37π180Suplementario:  π−53π180=127π180Coterminal:  2π+53π180=413π18053π180=53π180⋅180∘π=53∘\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=35y=\cfrac{3}{5}
  2. y=2425y=\cfrac{24}{25}
  1. H=2∧α=30∘H=2\wedge \alpha=30^{\circ}
  2. CA=3∧α=30∘CA=\sqrt{3}\wedge \alpha=30^{\circ}
  1. CO=1∧α=45∘CO=1\wedge \alpha=45^{\circ}
  2. H=5∧α=37∘H=5\wedge \alpha=37^{\circ}

Ejercicios 17 al 24.

  1. α=30.96∘,β=45∘,CE=5.83  pulgadas∧ED=7.07  pulgadas\alpha=30.96^{\circ} , \beta=45^{\circ},CE=5.83\; \text{pulgadas} \wedge ED=7.07\;\text{pulgadas}.
  2. α=δ=38.66∘,β=θ=γ=ρ=51.34∘,σ=77.32∘  y  CE=ED=6.4  pulgadas\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  metros de altura y una distancia de  939\; \text{metros de altura y una distancia de}\;9\sqrt{3}.
  4. $$$$
  1. a.  0.0103b.  0.0155\text{a.}\; 0.0103\\[0.1cm]\text{b.}\; 0.0155
  2. Altura (h)=1003m. Distancia(s)=300m.(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.14.9m., 17.3m.
  2. 10.3cm.,13.5cm.10.3cm., 13.5cm.
  3. 165.2m.165.2m.
  4. Son posibles todas comunicaciones, excepto entre Bernardo y Carlos.
  1. Juanita Martínez esta en lo correcto.
  2. 47.7  Km47.7\;Km
  3. a.  6.37 millas naˊuticasb.  16.80∘\text{a.\;6.37 millas náuticas}\\[0.1cm] \text{b.\;}16.80^{\circ}
  4. AC≈8468≈92AC \thickapprox\sqrt{8468}\thickapprox 92 pies.

Ejercicios 33 al 40.

  1. Sen2  θ(1+Cot2  θ)≡1Sen2θ(Csc2θ)≡Sen2θ(1Sen2θ)≡1≡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. Tanθ+Cotθ≡Sec  θ⋅Csc  θSen  θCos  θ+Cos  θSen  θ≡Sen2  θ+Cos2  θSen  θ⋅Cos  θ≡1Sen  θ⋅Cos  θ≡Sec  θ⋅Csc  θ≡Sec  θ⋅Csc  θ\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. Tan  θ+Cosθ1+Senθ≡Sec  θSen  θCos  θ+Cos  θ1+Sen  θ≡Sen  θ+Sen2  θ+Cos2  θCos  θ⋅(1+Sen  θ)≡Sen  θ+1Cos  θ⋅(1+Sen  θ)≡Sec  θ≡Sec  θ\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. 1+Sen  α1−Sen  α−1−Sen  α1+Sen  α≡4Tan  α⋅Sec  α(1+Sen  α)2−(1−Sen  α)2(1−Sen  α)⋅(1+Sen  α)≡1+2Sen  α+Sen2  α−1+2Sen  α−Sen2  α1−Sen2  α≡4Sen  αCos2  α≡4Sen  αCos  α⋅1Cos  α≡4Tan  α⋅Sec  α≡4Tan  α⋅Sec  α\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. Sen  θ(Csc  θ−Senθ)≡Cos2  θSen  θ(1Sen  θ−Senθ)≡Sen  θ(1−Sen2  θSen  θ)≡Cos2  θ≡Cos2  θ\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. (1−Cos2  θ)(1+Cot2θ)≡1(Sen2  θ)(Csc2θ)≡Sen2  θ(1Sen2  θ)≡1≡1\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. Cos  β(Tan  β+Cotβ)≡Csc  βCos  β(Sen  βCos  β+Cos  βSen  β)≡Cos  β(Sen2  β+Cos2  βCos  β⋅Sen  β)≡1Sen  β≡Csc  β≡Csc  β\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. Secθ−Tanθ≡1−Sen  θCos  θ1Cos  θ−Sen  θCos  θ≡1−Sen  θCos  θ≡1−Sen  θCos  θ\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. 7π6+NV(2π)\hspace{0.1cm}\cfrac{7\pi}{6}+NV\left(2\pi\right)
  2. π3+NV(2π),5π3+NV(2π),π+NV(2π)\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. π12+NV(π),5π12+NV(π)\hspace{0.1cm}\cfrac{\pi}{12}+NV\left(\pi\right),\cfrac{5\pi}{12}+NV\left(\pi\right)
  4. −π12+NV(2π),7π12+NV(2π)\hspace{0.1cm}\cfrac{-\pi}{12}+NV\left(2\pi\right),\cfrac{7\pi}{12}+NV\left(2\pi\right)
  1. π2+NV(π),2π3+NV(π),5π6+NV(π)\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. NV(360∘)\hspace{0.1cm}NV\left(360^{\circ}\right)
  3. 120∘+NV(360∘),240∘+NV(360∘)\hspace{0.1cm}120^{\circ}+NV\left(360^{\circ}\right), 240^{\circ}+NV\left(360^{\circ}\right)
  4. NV(π),2π3+NV(2π),4π3+NV(2π)\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. 4π3+NV(2π),5π3+NV(2π)\hspace{0.1cm}\cfrac{4\pi}{3}+NV\left(2\pi\right),\cfrac{5\pi}{3}+NV\left(2\pi\right)
  2. π4+NV(π),7π12+NV(π)\hspace{0.1cm}\cfrac{\pi}{4}+NV\left(\pi\right),\cfrac{7\pi}{12}+NV\left(\pi\right)
  3. NV(π)\hspace{0.1cm}NV\left(\pi\right)
  4. NV(2π),π2+NV(2π)\hspace{0.1cm}NV\left(2\pi\right),\cfrac{\pi}{2}+NV\left(2\pi\right)