Chapter 12, Problem 9SGR

To determine

To find: Solve $\Delta ABC$ .

Answer to Problem 9SGR

  $\begin{array}{l}c=15.5\\ a=4\\ A={15}^0\end{array}$

Explanation of Solution

Given information:

  $\begin{array}{l}b=15\\ B={75}^0\\ C={90}^0\end{array}$

Concept:

Law of sines

  $\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$

Given

  $\begin{array}{l}b=15\\ B={75}^0\\ C={90}^0\end{array}$

We know

Sum of angle of Triangle is ${180}^0$

Using that

We get

  $\begin{array}{l}A+B+C={180}^0\\ \Rightarrow A+{75}^0+{90}^0={180}^0\\ \Rightarrow A+{165}^0={180}^0\\ \Rightarrow A={180}^0-{165}^0\\ \Rightarrow A={15}^0\end{array}$

Using law of sines

We get

  $\begin{array}{l}\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}\\ \Rightarrow \frac{\sin {15}^0}{a}=\frac{\sin {75}^0}{15}=\frac{\sin {90}^0}{c}\end{array}$

Taking first two terms

We get

  $\begin{array}{l}\Rightarrow \frac{\sin {15}^0}{a}=\frac{\sin {75}^0}{15}\\ \Rightarrow a=\frac{15\sin {15}^0}{\sin {75}^0}\\ \Rightarrow a=\frac{15\left(0.259\right)}{0.966}\\ \Rightarrow a=\frac{3.885}{0.966}\\ \Rightarrow a=4\end{array}$

Using law of sines

Taking last two terms

We get

  $\begin{array}{l}\Rightarrow \frac{\sin {75}^0}{15}=\frac{\sin {90}^0}{c}\\ \Rightarrow c=\frac{15\sin {90}^0}{\sin {75}^0}\\ \Rightarrow c=\frac{15\left(1\right)}{0.966}\\ \Rightarrow c=\frac{15}{0.966}\\ \Rightarrow c=15.5\end{array}$

Therefore,

  $\begin{array}{l}c=15.5\\ a=4\\ A={15}^0\end{array}$