Compare Your Answers
  1. Solve for side \(h\) in the given triangle. Round to the nearest tenth of an inch.




    Step 1: Draw an altitude to make two right triangles.



    Step 2: Determine the height, \(t\), of the triangle.

    \(\theta = 35^\circ\)
    \({\mathop{\rm opp}\nolimits} = t\)
    \({\mathop{\rm hyp}\nolimits} = 8 \thinspace \rm{in}\)

    \(\begin{align}
     \sin \theta &= \frac{{{\mathop{\rm opp}\nolimits} }}{{{\mathop{\rm hyp}\nolimits} }} \\
     \sin 35^\circ &= \frac{t}{8} \\
     8\sin 35^\circ &= t \\
     4.588... &= t \\
     \end{align}\)

    Step 3: Determine the length of side \(h\).

    \(\theta = 15^\circ\)
    \({\mathop{\rm opp}\nolimits} = t = 4.588...\rm{in}\)
    \({\mathop{\rm hyp}\nolimits} = h\)

    \(\begin{align}
     \sin \theta &= \frac{{{\mathop{\rm opp}\nolimits} }}{{{\mathop{\rm hyp}\nolimits} }} \\
     \sin 15^\circ &= \frac{{4.588...}}{h} \\
     h &= \frac{{4.588...}}{{\sin 15^\circ }} \\
     h &= 17.729... \\
     h &\doteq 17.7 \\
     \end{align}\)

    Side \(h\) is approximately \(17.7\) in long.


  2. A cell tower has two guy wires attached at the same point. Naomi measures the angle of elevation from the wire closest to the tower to be \(43^\circ\) and the angle of elevation from the second wire to be \(32^\circ\). The distance between the two wires on the ground is \(6.3 \thinspace \rm{ft}\). How high up the tower are the wires connected? Round to the nearest foot.



    Step 1: Label unknown sides.



    Step 2: Solve for \(x\).

    \(\begin{align}
     \tan \theta &= \frac{{{\mathop{\rm opp}\nolimits} }}{{{\mathop{\rm adj}\nolimits} }} \\
     \tan 43^\circ &= \frac{y}{x} \\
     x\tan 43^\circ &= y \\
     \end{align}\)
    \(\begin{align}
     \tan \theta &= \frac{{{\mathop{\rm opp}\nolimits} }}{{{\mathop{\rm adj}\nolimits} }} \\
     \tan 32^\circ &= \frac{y}{{x + 6.3}} \\
     \left( {x + 6.3} \right)\tan 32^\circ &= y \\
     \end{align}\)

    Both \(y\)’s will be equal, therefore

    \(\begin{align}
     x\tan 43^\circ &= \left( {x + 6.3} \right)\tan 32^\circ  \\
     0.932...x &= 0.624...x + 3.936... \\
     0.307...x &= 3.936... \\
     x &= 12.796... \\
     \end{align}\)

    Step 3: Solve for \(y\).

    \(\begin{align}
     y &= x\tan 43^\circ  \\
     y &= \left( {12.796...} \right)\tan 43^\circ  \\
     y &= 11.932... \\
     y &\doteq 12\thinspace {\rm{ ft}} \\
     \end{align}\)


    The wires are attached at a height of approximately \(12\) ft up on the tower.