Chapter 6.2, Problem 12E

To determine

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

Answer to Problem 12E

The value of length of side x of the right angled triangle is $12\sqrt{2}$ 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 12 units and hypotenuse has length x 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 $\sin 45°=\frac{1}{\sqrt{2}}$ .

Apply it, to estimate the value of trigonometric ratios,

The value of sine function is,

  $\begin{array}{c}\sin 45°=\frac{\text{opposite}}{\text{hypotenuse}}\\ \frac{1}{\sqrt{2}}=\frac{12}{x}\end{array}$

Multiply each side by x ,

  $\begin{array}{c}\frac{1}{\sqrt{2}}\cdot x=\frac{12}{x}\cdot x\\ \frac{x}{\sqrt{2}}=12\end{array}$

Multiply each side by $\sqrt{2}$ ,

  $\begin{array}{c}\frac{1}{\sqrt{2}}\cdot x=\frac{12}{x}\cdot x\\ \frac{x}{\sqrt{2}}\cdot \sqrt{2}=12\cdot \sqrt{2}\\ x=12\sqrt{2}\end{array}$

Thus, the value of length of side x of the right angled triangle is $12\sqrt{2}$ units.