Chapter 12.1, Problem 53PPS

To determine

Calculate the missing length and the missing angles of the triangle ABC.

Answer to Problem 53PPS

The angles are ${38.7}^{\circ }$ & ${51.3}^{\circ }$ and the sides are $7.5$ & $9.6$ .

Explanation of Solution

Given:

It is given in the question that the $\tan A=\frac{4}{5}\&a=6$ and $\angle C={90}^{\circ }$ .

Concept Used:

In this, use the concept of angle sum property and of trigonometric ratios and the formula will be, $\sin \theta =\frac{p}{h},\cos \theta =\frac{b}{h},\tan \theta =\frac{p}{b},\csc \theta =\frac{h}{p},\sec \theta =\frac{h}{b},\cot \theta =\frac{b}{p}$ and may also use the pythgoras theorem.

Calculation:

Now, using trigonometric ratios,

  $\begin{array}{l}\tan \theta =\frac{p}{b}\\ \tan A=\frac{4}{5}\\ A={\tan }^{-1}(\frac{4}{5})\\ A={38.7}^{\circ }\end{array}$

Solve for b,

  $\begin{array}{l}\frac{4}{5}=\frac{6}{b}(\tan \theta =\tan A)\\ 4b=30\\ b=7.5\end{array}$

Now, use pythagoras theorem,

  $\begin{array}{l}{a}^2+b^2=c^2\\ 6^2+{7.5}^2=c^2\\ c^2=36+56.25\\ c^2=92.25\\ c=\sqrt{92.25}=9.6\end{array}$

Now from angle sum property,

  $\begin{array}{l}{38.7}^{\circ }+{90}^{\circ }+\angle B=180\\ \angle B=180-(38.7+90)\\ \angle B=180-128.7\\ \angle B={51.3}^{\circ }\end{array}$

Conclusion:

The angles are ${38.7}^{\circ }$ & ${51.3}^{\circ }$ and the sides are $7.5$ & $9.6$ .