Chapter 13.1, Problem 38WE

To determine

To find: The all to three significant digits and angles.

Answer to Problem 38WE

The angle is $\angle A={33.2}^o$ , $\angle C={42.8}^o$ and side is $a\approx 5.67$ .

Explanation of Solution

Given information:

The given angle $\angle B={104}^o$ , and sides $b=12$ and $c=7$ .

Calculation:

Consider the triangle.

  $\angle A={75}^o$ , $\angle B={30}^o$ , $b=12$ .

  $\begin{array}{l}C={180}^o-\left(75+30\right)\\ C=75\end{array}$

Now, use the law of sine to find angle $C$ .

  $\frac{\sin B}{b}=\frac{\sin C}{c}$

Substitute the value of $b,c$ and $B$ .

  $\begin{array}{c}\frac{\sin {104}^o}{10}=\frac{\sin C}{7}\\ \frac{7\sin {104}^o}{10}=\sin C\\ \sin C=0.68\end{array}$

Use the inverse sine function.

  $\begin{array}{c}C={\sin }^{-1}\left(0.68\right)\\ ={42.8}^o\end{array}$

Since the sum of all three angles is ${180}^o$ .

  $\begin{array}{c}\angle A+\angle B+\angle C={180}^o\\ \angle A+{104}^o+{42.8}^o={180}^o\\ \angle A={180}^o-{146.8}^o\\ ={33.2}^o\end{array}$

Again,

Use law of sine to find the length of $a$ .

  $\frac{\sin A}{a}=\frac{\sin B}{b}$

Substitute the value of $B,A$ and $b$ .

  $\begin{array}{c}\frac{\sin {33.2}^o}{a}=\frac{\sin {104}^o}{10}\\ \frac{\sin {33.2}^o}{a}\approx 0.097\end{array}$

Multiply by $a$ both sides.

  $\begin{array}{c}a\left(\frac{\sin {33.2}^o}{a}\right)=a\left(0.097\right)\\ \sin {33.2}^o\approx 0.097a\\ 0.55\approx 0.097a\\ a\approx 5.67\end{array}$

Therefore, the angle is $\angle A={33.2}^o$ , $\angle C={42.8}^o$ and side is $a\approx 5.67$ .