Chapter 12.5, Problem 14P

To determine

Find x .

Answer to Problem 14P

About 5 cm.

Explanation of Solution

Given:

  

Formula Used:

  $\sin \theta =\frac{\text{length of opposite side to }\theta }{\text{length of hypotenuse }}$

Calculation:

Extend the triangle :

  

In the largest triangle , find the hypotenuse ,

  $\begin{array}{l}\cos 26°=\frac{\text{length of adjacent side to 26}°}{\text{length of hypotenuse }}\\ \Rightarrow \cos 26°=\frac{50}{hypotenuse}\\ \Rightarrow hypotenuse=\frac{50}{\cos 26°}\mathrm{...................}[\text{multiply each side by }\frac{hypotenuse}{\cos 26°}\text{]}\\ \Rightarrow hypotenuse=55.63\end{array}$

So, the hypotenuse of the smaller right angled triangle = 55.63-50 = 5.63 cm

In the same triangle ,

  $\begin{array}{l}\sin \theta =\frac{\text{length of opposite side to }\theta }{\text{length of hypotenuse }}\\ \Rightarrow \sin \theta =\frac{50}{55.63}\\ \Rightarrow \theta ={\sin }^{-1}\mathrm{0.898...................}[\text{multiply each side by a]}\\ \Rightarrow \theta \approx 64°\end{array}$

In the smaller triangle ,

  $\begin{array}{l}\sin 64°=\frac{\text{length of opposite side to 64}°}{\text{length of hypotenuse }}\\ \Rightarrow \sin 64°=\frac{x}{5.63}\\ \Rightarrow x=5.63\sin 64°\mathrm{...................}[\text{multiply each side by 5}\text{.63]}\\ \Rightarrow x=5.63(0.89879)\\ \Rightarrow x=5.0601877\end{array}$

So, x is about 5 cm.