Chapter 13, Problem 8SGR

To determine

To find: Solve $\Delta ABC$ .

Answer to Problem 8SGR

  $\begin{array}{l}b=14.3\\ c=17.4\\ A={35}^0\end{array}$

Explanation of Solution

Given information:

  $\begin{array}{l}a=10\\ B={55}^0\\ C={90}^0\end{array}$

Concept:

Law of sines

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

Given

  $\begin{array}{l}a=10\\ B={55}^0\\ C={90}^0\end{array}$

We know

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

Using that

We get

  $\begin{array}{l}A+B+C={180}^0\\ \Rightarrow A+{55}^0+{90}^0={180}^0\\ \Rightarrow A+{145}^0={180}^0\\ \Rightarrow A={180}^0-{145}^0\\ \Rightarrow A={35}^0\end{array}$

Using law of sines

We get

  $\begin{array}{l}\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}\\ \Rightarrow \frac{\sin {35}^0}{10}=\frac{\sin {55}^0}{b}=\frac{\sin {90}^0}{c}\end{array}$

Taking first and last terms

We get

  $\begin{array}{l}\Rightarrow \frac{\sin {35}^0}{10}=\frac{\sin {90}^0}{c}\\ \Rightarrow c=\frac{10\sin {90}^0}{\sin {35}^0}\\ \Rightarrow c=\frac{10\left(1\right)}{0.5736}\\ \Rightarrow c=17.4\end{array}$

Using law of sine

Taking first two terms

We get

  $\begin{array}{l}\Rightarrow \frac{\sin {35}^0}{10}=\frac{\sin {55}^0}{b}\\ \Rightarrow b=\frac{10\sin {55}^0}{\sin {35}^0}\\ \Rightarrow b=\frac{10\left(0.82\right)}{0.5736}\\ \Rightarrow b=\frac{8.2}{0.5736}\\ \Rightarrow b=14.3\end{array}$

Therefore,

  $\begin{array}{l}b=14.3\\ c=17.4\\ A={35}^0\end{array}$