Chapter 9.5, Problem 99E

To determine

To calculate:

The values of the triangle by using the sine or cosine law:

  $B=71°,a=21,c=29$

Answer to Problem 99E

The values of the triangle are:

  $\begin{array}{l}A=41.85°\\ C=67.15°\\ b=29.76\end{array}$

Explanation of Solution

Given information:

  $B=71°,a=21,c=29$

Formula used:

The cosine law to solve the triangle is given by the following expressions:

  $\begin{array}{l}\cos A=\frac{b^2+c^2-a^2}{2bc}\\ \cos B=\frac{c^2+b^2-b^2}{2ca}\\ \cos C=\frac{a^2+b^2-c^2}{2ab}\end{array}$

The sine law to solve the triangle is given by the following expressions:

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

Calculation:

The value of $b$ is estimated as:

  $\begin{array}{l}\cos B=\frac{c^2+b^2-b^2}{2ca}\\ \cos 71=\frac{{21}^2+{29}^2-b^2}{2\left(21\right)\left(29\right)}\\ b^2=1282-396.581\\ b^2=885.419\\ b=29.76\end{array}$

To construct the triangle, need to find other values by using the sine law:

For $A$

  $\begin{array}{l}\frac{a}{\sin A}=\frac{b}{\sin B}\\ \frac{21}{\sin A}=\frac{29.76}{\sin 71°}\\ A={\sin }^{-1}\left(0.6672\right)\\ A=41.85°\end{array}$

The third angle of the triangle is estimated by using the angle theorem:

  $\begin{array}{l}A+B+C=180\\ C=180-\left(A+B\right)\\ C=180-\left(41.85+71\right)\\ C=67.15°\end{array}$