Chapter 8.6, Problem 13WE

a.

To determine

To find: the value of x in radical form.

Answer to Problem 13WE

The value of x is $\sqrt{115}$

Explanation of Solution

Given Information: A right angled triangle with hypotenuse and altitude given as 14 and 9 units respectively. The acute angle and adjacent side is shown as y⁰and x respectively.

Formula used:

The relationship between the legs of right triangle (shown in figure below) a , b and its hypotenuse c is obtained by using Pythagoras theorem.

  

It can be expressed as,

  $\begin{array}{l}{c}^2=a^2+b^2\\ c=\sqrt{a^2+b^2}\end{array}$

Calculation: Consider the triangle given below.

  

Applying Pythagoras theorem, the value of x which represents the adjacent side can be calculated as,

  $\begin{array}{c}x=\sqrt{{14}^2-9^2}\\ =\sqrt{115}\end{array}$

b.

To determine

To find: the value of y using trigonometry.

Also, to find the value of x using y

Answer to Problem 13WE

The measure of y is $\simeq \text{6 units}$

Explanation of Solution

Given Information: A right angled triangle with hypotenuse and altitude given as 14 and 9 units respectively. The acute angle and adjacent side is shown as y⁰and x respectively.

Formula used:

  

In a right triangle, the sine of an angle is the ratio of opposite side to the hypotenuse of the triangle expressed as,

  $\sin B=\frac{a}{c}$

Also, the tangent of an angle is the ratio of opposite side to the adjacent side of the triangle expressed as,

  $\tan B=\frac{a}{b}$

Calculation: Consider the triangle given below. The sine of angle y can be expressed as,

  

  $\begin{array}{c}\sin y°=\frac{9}{14}\\ =0.6429\end{array}$ .

Therefore, angle y can be calculated as,

  $\begin{array}{c}y°={\sin }^{-1}\left(0.6429\right)\\ =40.005°\\ \simeq 40°\end{array}$

Now, the value of x can be calculated by considering tangent of angle y⁰.

  $\begin{array}{c}\tan y°=\frac{9}{x}\\ \tan 40°=\frac{9}{x}\end{array}$

Rearranging the equation yields the value of x as,

  $\begin{array}{c}x=\frac{9}{\tan 40°}\\ =\frac{9}{0.8393}\\ =10.7238\end{array}$

c.

To determine

To check: if the value of x in parts a. and b. are in reasonable agreement.

Answer to Problem 13WE

Yes

Explanation of Solution

Given Information: A right angled triangle with hypotenuse and altitude given as 14 and 9 units respectively. The acute angle and adjacent side is shown as y⁰and x respectively. The value of x in both parts a. and b. are to be considered.

The value of x calculated in part a. using the Pythagoras theorem is,

  $\begin{array}{c}x=\sqrt{115}\\ =10.7238\end{array}$

Now, the value of x found in part b. using trigonometry relations is,

  $x=10.7238$

It can be seen that, the values are in reasonable agreement with each other and are exactly the same in both cases.