Chapter 6.3, Problem 57E

To determine

To calculate: The angle included by the two sides of the triangle.

Answer to Problem 57E

The angle included by the two sides of the triangleis $66.1°$ .

Explanation of Solution

Given information:

A triangle with length of its two sides as $5\text{ in}\text{.}$ and $7\text{ in}$ . Also the area of the triangle is $16{\text{ in}}^2$ .

Formula used:

The area of a triangle is denoted by $A=\frac{1}{2}ab\sin \theta$ where a andb are the length of its sides and $\theta$ is the included angle.

Calculation:

Consider the equilateral triangle such that length of its sides is 10.triangle with length of its two sides as $5\text{ in}\text{.}$ and $7\text{ in}$ . Also the area of the triangle is $16{\text{ in}}^2$ .

Recall that the area of a triangle is denoted by $A=\frac{1}{2}ab\sin \theta$ where a andb are the length of its sides and $\theta$ is the included angle.

Apply it,

  $\begin{array}{c}16=\frac{1}{2}\left(5\right)\left(7\right)\sin \theta \\ 32=35\sin \theta \\ \sin \theta =\frac{32}{35}\\ \sin \theta =0.9142\end{array}$

Simplify it further as,

  $\begin{array}{c}\theta ={\sin }^{-1}\left(0.9142\right)\\ \theta =66.1°\end{array}$

Thus, the angle included by the two sides of the triangle is $66.1°$ .