Chapter 12.5, Problem 14WE

To determine

Solve the right triangle ABC for which $\angle B=67°20\text{'}\text{ and }a=450$ . Give the length to three significant digits and angle measures to nearest ten minutes.

  

Answer to Problem 14WE

The three sides of triangle are $a=450,b=1077.520\text{ and }c=1167.711$ . And three angles are $\angle A=22°40\text{'},\angle B=67°20\text{ and }\angle C=90.0°$ .

Explanation of Solution

Given information: The given triangle ABC has $\angle B=67°20\text{'}\text{ and }a=450$ .

  

Calculation: Since the triangle ABC has $\angle B=67°20\text{'}\text{ and }a=450$ . And it is right angle at point $\left(C\right)$ . Therefore in triangle ABC,

  $\begin{array}{c}\angle A=90-67°20\text{'}\\ \angle A=22°40\text{'}\end{array}$

Therefore by trigonometric ratios,

  $\begin{array}{c}\sin A=\frac{a}{c}\\ \sin \left(22°40\text{'}\right)=\frac{450}{c}\\ \text{or, }c=\frac{450}{\sin \left(22+\frac{40}{60}\right)}\\ c\approx 1167.7111\end{array}$

Similarly,

  $\begin{array}{c}\tan A=\frac{a}{b}\\ \tan \left(22°40\text{'}\right)=\frac{450}{b}\\ \text{or, }b=\frac{450}{\tan \left(22+\frac{40}{60}\right)}\\ b\approx 1077.520\end{array}$

Hence the given triangle ABC has sides $a=450,b=1077.520\text{ and }c=1167.711$ . And three angles are $\angle A=22°40\text{'},\angle B=67°20\text{ and }\angle C=90.0°$ .