Question

Solve triangle ABC using the given measurements if $B=56{}^{\circ },C=90{}^{\circ }\text{and}c=16$. Round measure of sides to the nearest tenth and angles to nearest degree.

Short answer

The solution for triangle ABC is$a=8.9,b=13.3\text{and}A=34°$.

Step-by-step solution

Step 1. Write down the given parameters of triangle.

The given triangle ABC is shown below whose parameters are represented as $B=56{}^{\circ },C=90{}^{\circ }\text{and}c=16$.

Step 2. Calculation.

In triangle ABC, $B=56{}^{\circ },C=90{}^{\circ }\text{and}c=16$. Therefore, $\angle A=34°$ by angle sum property.

Now, trigonometric ratios,

$\begin{array}{c}\sin B=\frac{b}{c}\\ \sin 56°=\frac{b}{16}\\ b=16\sin \left(56°\right)\\ b=13.3\end{array}$

Now, in right triangle ABC,

$\begin{array}{c}{c}^2=a^2+b^2\text{\hspace{0.17em}gives,}\\ {16}^2=a^2+{13.3}^2\\ a^2={16}^2-{13.3}^2\\ a=\sqrt{{16}^2-{13.3}^2}\\ a=8.9\end{array}$

Step 3. Conclusion.

Thesolution for triangle ABC, is $a=8.9,b=13.3\text{and}A=34°$.