Chapter 4.8, Problem 7E

To determine

To solve: The right triangle.

Answer to Problem 7E

The solution of the triangle is ${\text{A = 19}}^{\circ },{\text{ B = 71}}^{\circ },{\text{ C = 90}}^{\circ }\text{ and a = 4}\text{.83, b =14, c = 14}\text{.81}$.

Explanation of Solution

Given information:

  

$\text{B}={71}^{\circ },\text{ b}=14$

Calculation: From the properties of trigonometric function,

  $\begin{array}{l}\sin \theta =\frac{\text{Perpendicular}}{\text{Hypotenuse}},\text{ csc}\theta \text{=}\frac{\text{Hypotenuse}}{\text{Perpendicular}}\\ \cos \theta =\frac{\text{Base}}{\text{Hypotenuse}},\text{ sec}\theta \text{=}\frac{\text{Hypotenuse}}{\text{Base}}\\ \tan \theta =\frac{\text{Perpendicular}}{\text{Base}},\text{ cot}\theta =\frac{\text{Base}}{\text{Perpendicular}}\end{array}$   

From the triangle,

  $\text{B}={71}^{\circ },\text{ C}={90}^{\circ }$

Therefore,

  $\begin{array}{c}\text{A+B+C}={180}^{\circ }\\ \text{A}+{71}^{\circ }+{90}^{\circ }={180}^{\circ }\\ \text{A}={180}^{\circ }-{90}^{\circ }-{71}^{\circ }\\ \text{A}={19}^{\circ }\end{array}$

Now, $\text{a}=\text{Perpendicular, b = Base, c = Hypotenuse}$

Since, $\text{b}=14$

From the properties of a right triangle,

  $\begin{array}{c}\cos {19}^{\circ }=\frac{\text{b}}{\text{c}}\\ 0.9455=\frac{14}{\text{c}}\\ \text{c}=\frac{14}{0.9455},\text{ doing cross-multiplication}\\ \text{c = 14}\text{.81 }\end{array}$

Therefore, Base = $14$ and Hypotenuse = $14.81$

As per Pythagoras Theorem,

  $\begin{array}{c}\text{ Hypotenuse}=\sqrt{{\left(\text{Perpendicular}\right)}^2+{\left(\text{Base}\right)}^2}\\ 14.81=\sqrt{{\left(\text{Perpendicular}\right)}^2+{\left(14\right)}^2}\\ 219.336={\text{Perpendicular}}^2+196\\ \text{Perpendicular}=\sqrt{219.336-196}\\ \text{Perpendicular}=\sqrt{23.336}\\ \text{Perpendicular}=4.83\end{array}$

Hence, The third side of the triangle is, perpendicular $=4.83$

Hence, The solution of the triangle is ${\text{A = 19}}^{\circ },{\text{ B = 71}}^{\circ },{\text{ C = 90}}^{\circ }\text{ and a = 4}\text{.83, b =14, c = 14}\text{.81}$