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Completion requirements
- Determine the value of \(r\), to the nearest tenth.
Begin by labelling the diagram.
Now, use the appropriate sine law equation.
\(\begin{align}
\frac{a}{{\sin A}} &= \frac{b}{{\sin B}} \\
\frac{r}{{\sin 37^\circ }} &= \frac{{51}}{{\sin 112^\circ }} \\
r &= \frac{{51\sin 37^\circ }}{{\sin 112^\circ }} \\
r &= 33.102... \\
r &\doteq 33.1 \\
\end{align}\) - In
his garage, Gerald is hoisting an engine from a pickup truck he is
fixing. He hangs the hoist using three rafters on the garage ceiling. He
measures the angle the hoist makes at the base to be \(85^\circ\), and
he knows the length of chain across the rafters is \(80\) in. He
measures one length of chain hanging down from the rafter to the hoist
to be \(60\) in. What is the length, to the nearest tenth of an
inch, of the other section of chain?
What is given?
\(\angle A = 85^\circ\)
\(a = 80\thinspace \rm{in}\)
\(b = 60\thinspace \rm{in}\)
\(c = \thinspace ?\)
Because the known angle is not between the known sides, you cannot use the cosine law. Using the sine law will require two steps.
Step 1: Determine \(\angle B\) and \(\angle C\).
\(\begin{align}
\frac{a}{{\sin A}} &= \frac{b}{{\sin B}} \\
\frac{{80}}{{\sin 85^\circ }} &= \frac{{60}}{{\sin B}} \\
\sin B &= \frac{{60\sin 85^\circ }}{{80}} \\
\sin B &= 0.747... \\
\angle B &= \sin ^{ - 1} \left( {0.747...} \right) \\
\angle B &= 48.343...^\circ \\
\end{align}\)
\(\begin{align}
\angle C&= 180^\circ - 85^\circ - 48.343...^\circ \\
&= 46.656...^\circ \\
\end{align}\)
Step 2: Determine side \(c\).
\(\begin{align}
\frac{a}{{\sin A}} &= \frac{c}{{\sin C}} \\
\frac{{80}}{{\sin 85^\circ }} &= \frac{c}{{\sin 46.656...^\circ }} \\
\frac{{80\sin 46.656...^\circ }}{{\sin 85^\circ }} &= c \\
58.402... &= c \\
58.4 &\doteq c \\
\end{align}\)
The length of the other section of chain is approximately \(58.4\) in.
| For further information about the sine law see pp. 102 to 104 of Pre-Calculus 11. |