Chapter 3, Problem 1RE

To determine

To calculate: The value of angle $C$ and sides $b,c$ with the help of the law of sines for the triangle $ABC$ as shown below, and also round the result to two decimal places.

Answer to Problem 1RE

The value of the angle $C\text{ is }72°$ and the length of sides $b\text{ and }c$ are approximately $12.21$ and $12.35$, respectively.

Explanation of Solution

Calculation:

Consider the given triangle $ABC$,

Observe from the triangle $ABC$, the angles, $A\text{is}\text{}38°$, $B\text{is}70°$ and the side $a\text{ is }28$.

Now, calculate the remaining angle $C$ of the triangle $ABC$,

$C=180°-A-B$

Where, $180°$ is the sum of the three angles of the triangle $ABC$.

Substitute the value of $A\text{ and }B$ in the formula $C=180°-A-B$,

$\begin{array}{c}C=180°-A-B\\ =180°-38°-70°\\ =72°\end{array}$

Now, calculate the side $b$ and $c$ of the triangle $ABC$,

Calculate the side $b\text{ and }c$ by the law of sines,

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

Consider the first two terms of the law of sines, $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$,

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

Multiply by $\sin B$ on both sides of the equation, $\frac{a}{\sin A}=\frac{b}{\sin B}$.

$\begin{array}{c}\frac{a}{\sin A}=\frac{b}{\sin B}\\ b=\frac{a}{\sin A}\cdot \sin B\end{array}$

Substitute the values of $a\text{ and }A,B$ in the equation, $b=\frac{a}{\sin A}\cdot \sin B$.

$\begin{array}{c}b=\frac{a}{\sin A}\cdot \sin B\\ =\frac{8\sin 70°}{\sin 38°}\\ =\frac{8\cdot \left(0.939\right)}{0.615}\\ \approx 12.21\end{array}$

Now, consider the first and last term of the law of sines, $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$, to get the value of $c$,

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

Multiply by $\sin C$ on both sides of the equation, $\frac{a}{\sin A}=\frac{c}{\sin C}$.

$\begin{array}{c}\frac{a}{\sin A}=\frac{c}{\sin C}\\ c=\frac{a}{\sin A}\cdot \sin C\end{array}$

Substitute the values of $a,C,A$ in the equation, $c=\frac{a}{\sin A}\cdot \sin C$.

$\begin{array}{c}c=\frac{a}{\sin A}\cdot \sin C\\ =\frac{8}{\sin 38°}\cdot \sin 72°\\ =\frac{8\cdot \left(0.95105\right)}{0.61566}\\ \approx 12.35\end{array}$

Therefore, the value of the angle $C\text{ is }72°$ and the length of sides $b\text{ and }c$ are approximately $12.21$ and $12.35$, respectively.