Chapter 6.2, Problem 13E

To determine

To calculate: The value of length of side x of the right angled triangle.

Answer to Problem 13E

The value of length of side x of the right angled triangle is $11.25833$ units.

Explanation of Solution

Given information:

The right angle triangle with length of its sides and angle.

Formula used:

The trigonometric ratios for a right angle triangle are defined as,

  $\sin \theta =\frac{\text{opposite}}{\text{hypotenuse}},cos\theta =\frac{\text{adjacent}}{\text{hypotenuse}},\tan \theta =\frac{\text{opposite}}{\text{adjacent}},\csc \theta =\frac{\text{hypotenuse}}{\text{opposite}},\sec \theta =\frac{\text{hypotenuse}}{\text{adjacent}}$ and $\cot \theta =\frac{\text{adjacent}}{\text{opposite}}$ .

Calculation:

Consider the right angle triangle with length of its sides and angle.

Observe that opposite side is of length x units and hypotenuse has length 13 units.

Recall that the sine trigonometric ratio for a right angle triangle is defined as, $\sin \theta =\frac{\text{opposite}}{\text{hypotenuse}}$ .

The value of trigonometric function $\sin 60°=\frac{\sqrt{3}}{2}$ .

Apply it, to estimate the value of trigonometric ratios,

The value of sine function is,

  $\begin{array}{c}\sin 60°=\frac{\text{opposite}}{\text{hypotenuse}}\\ \frac{\sqrt{3}}{2}=\frac{x}{13}\end{array}$

Multiply each side by 13.

  $\begin{array}{c}\frac{\sqrt{3}}{2}\cdot 13=\frac{x}{13}\cdot 13\\ x=\frac{13\sqrt{3}}{2}\end{array}$

Simplify the value with help of calculator,

  $x=11.25833$

Thus, the value of length of side x of the right angled triangle is $11.25833$ units.