Example  3
 

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A surveyor was working in a field between three points that form an obtuse triangle. She found the distances between the points, on either side of the obtuse angle, to be \(161 \thinspace \rm{m}\) and \(362 \thinspace \rm{m}\). She found the acute angle adjacent to the \(362 \thinspace \rm{m}\) distance to be \(21^\circ\).

  1. What is the measure of the obtuse angle, to the nearest tenth of a degree?

    Begin by drawing a diagram to represent the situation.



    The goal is to determine the value of \(\angle B\). Unfortunately, you are not given side \(b\), so you cannot use the sine law directly to determine \(\angle B\). However, it is possible to determine \(\angle A\), and then use the sum of the interior angles in a triangle to determine \(\angle B\).

    \(\begin{align}
     \frac{{\sin A}}{a} &= \frac{{\sin C}}{c} \\
     \frac{{\sin A}}{{362}} &= \frac{{\sin 21^\circ }}{{161}} \\
     \sin A &= \frac{{362\sin 21^\circ }}{{161}} \\
     \sin A &= 0.805... \\
     \angle A &= \sin ^{ - 1} \left( {0.805...} \right) \\
     \angle A &= 53.684...^\circ  \\
     \angle A &\doteq 53.7^\circ  \\
     \end{align}\)

    Determine \(\angle B\), using the sum of the interior angles in a triangle.

    \(\begin{align}
     \angle B &= 180^\circ - 53.684...^\circ - 21^\circ \\
     \angle B &= 105.315...^\circ  \\
     \angle B &\doteq 105.3^\circ  \\
     \end{align}\)

  2. What should the surveyor do if the angle she calculates does not match the measured angle?

    If the measured angle and the calculated angle don’t match, there is an error somewhere. The surveyor will need to recheck her measurements and calculations, and correct any errors until the two agree.