Q39. - Algebra 2

Question

Solve triangle ABC using the given measurements if $\tan B=\frac{7}{8},C=90{}^{\circ }\text{ and } b=7$. Round measure of sides to the nearest tenth and measures of angles to nearest degree.

Step-by-Step Solution

Answer

The solution for triangle ABC is $c=10.6,a=8,\angle A=49{}^{\circ }\text{ and }\angle B=41°$.

Step 1. Write down the given parameters of triangle.

The given triangle ABC is shown below whose parameters are represented as $\tan B=\frac{7}{8},C=90{}^{\circ }\text{ and } b=7$. 

Step 2. Calculation.

In triangle ABC, $\tan B=\frac{7}{8},C=90{}^{\circ }\text{ and } b=7$.

Therefore, $\frac{b}{a}=\frac{7}{8}$. Plugging for $b=7$, gives $a=8$.

Now, in right triangle ABC,

 $\begin{array}{c}{c}^2=a^2+b^2\text{\hspace{0.17em}gives,}\\ c^2=8^2+7^2\\ c=\sqrt{8^2+7^2}\\ c=10.6\end{array}$

Now by trigonometric ratios,

$\begin{array}{c}\tan B=\frac{7}{8}\\ B={\tan }^{-1}\left(\frac{7}{8}\right)\\ B=41°\end{array}$ 

Now, $C=90{}^{\circ }\text{ and }B=41°$, therefore by angle sum property $\angle A=49°$.

Step 3. Conclusion.

The solution for triangle ABC, is $c=10.6,a=8,\angle A=49{}^{\circ }\text{ and }\angle B=41°$.