Question

Solve triangle ABC using the given measurements if $A=16{}^{\circ },C=90{}^{\circ }\text{and}c=14$. Round measure of sides to the nearest tenth and measures of angles to nearest degree.

Short answer

The solution for triangle ABC is$a=13.4,b=3.9\text{and}B=74°$.

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 $A=16{}^{\circ },C=90{}^{\circ }\text{and}c=14$.

Step 2. Calculation.

In triangle ABC, $A=16{}^{\circ },C=90{}^{\circ }\text{and}c=14$. Therefore, $\angle B=74°$by angle sum property.

Now, trigonometric ratios,

$\begin{array}{c}\sin B=\frac{b}{c}\\ \sin 16°=\frac{b}{14}\\ b=14\sin \left(16°\right)\\ b=3.9\end{array}$

Now, in right triangle ABC by Pythagoras Theorem.

$\begin{array}{c}{c}^2=a^2+b^2\text{\hspace{0.17em}gives,}\\ {14}^2=a^2+{3.9}^2\\ a^2={14}^2-{3.9}^2\\ a=\sqrt{{14}^2-{3.9}^2}\\ a=13.4\end{array}$

Step 3. Conclusion.

Thesolution for triangle ABC, is $a=13.4,b=3.9\text{and}B=74°$.