Chapter 10.6, Problem 33P

For each right triangle described, find all three angles to the nearest tenth.

The hypotenuse is 8 ft long. Hie adjacent side is 5 ft long.

To determine

To find: All three angles of a right angled triangle and round to the nearest tenth

Answer to Problem 33P

The values of all the three angles of a right angled triangle are, $\angle A={39.7}^{\circ }$ , $\angle B={90}^{\circ }$ and $\angle C={51.3}^{\circ }$

Explanation of Solution

Given information:

The hypotenuse is 8 feet long and the adjacent side is 5 feet long

Formula used:

  $\cos \theta =\frac{\text{Base}}{\text{Hypotenuse}}$

Calculation:

For a right angled triangle, assume hypotenuse as $AC$ , base as $BC$ and perpendicular as $AB$

The length of hypotenuse $\left(AC\right)$$=8\text{ feet}$

The adjacent side $\left(BC\right)=5\text{ feet}$

Now,

  $\begin{array}{l}\cos \theta =\frac{\text{Base}}{\text{Hypotenuse}}\\ \cos \theta =\frac{BC}{AC}\\ \cos \theta =\frac{5}{8}\\ \cos \theta =0.625\\ \theta ={\cos }^{-1}\left(0.625\right)\\ \theta ={51.34}^{\circ }\end{array}$

Hence, $\angle C=\theta ={51.34}^{\circ }={51.3}^{\circ }\left(\text{Rounded to nearest tenth}\right)$ and $\angle B={90}^{\circ }$ as it is a right angled triangle

The values of all the three angles of a right angled triangle can be calculated as,

  $\begin{array}{l}\angle A+\angle B+\angle C={180}^{\circ }\left[{\text{Sum of angles of triangle is 180}}^{\circ }\right]\\ \angle A={180}^{\circ }-\left(\angle B+\angle C\right)\\ \angle A={180}^{\circ }-\left({90}^{\circ }+{51.34}^{\circ }\right)\\ \angle A={39.66}^{\circ }\\ \angle A={39.7}^{\circ }\left[\text{Rounded to nearest tenth}\right]\end{array}$

Therefore, the values of all the three angles of a right angled triangle are, $\angle A={39.7}^{\circ }$ , $\angle B={90}^{\circ }$ and $\angle C={51.3}^{\circ }$