Chapter 6.2, Problem 32E

To determine

To calculate: The sides of the right angled triangle.

Answer to Problem 32E

The sides of right angled triangle are $100,26.79\text{ and }103.53$ . Triangle so formed is provided below,

Explanation of Solution

Given information:

The right angle triangle with length of its sides and angle $\theta$ .

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 $\theta$ .

Observe that the length of opposite side is 16 units.

Label the length of adjacent side of the triangle as x and hypotenuse as y .

Recall that 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 $\cot \theta =\frac{\text{adjacent}}{\text{opposite}}$ .

Apply it, to estimate the value of trigonometric ratios,

The value of sine function is,

  $\begin{array}{c}\sin 75°=\frac{\text{opposite}}{\text{hypotenuse}}\\ \sin 75°=\frac{100}{y}\end{array}$

Multiply both sides by y ,

  $\begin{array}{c}\sin 75°\cdot y=\frac{100}{y}\cdot y\\ \sin 75°\cdot y=100\end{array}$

Multiply both sides by $\sin 75°$ ,

  $\begin{array}{c}\sin 75°\cdot y=\frac{100}{y}\cdot y\\ \frac{\sin 75°\cdot y}{\sin 75°}=\frac{100}{\sin 75°}\\ y=\frac{100}{\sin 75°}\end{array}$

Simplify it with help of calculator,

  $\begin{array}{c}\sin 75°\cdot y=\frac{100}{y}\cdot y\\ \frac{\sin 75°\cdot y}{\sin 75°}=\frac{100}{\sin 75°}\\ y=\frac{100}{\sin 75°}\\ y=103.53\end{array}$

Therefore, the length of hypotenuse is $103.53$ units.

The value of cosine function is,

  $\begin{array}{c}cos75°=\frac{\text{adjacent}}{\text{hypotenuse}}\\ cos75°=\frac{x}{103.53}\end{array}$

Multiply both sides by $103.53$ ,

  $\begin{array}{c}103.53\cdot cos75°=103.53\cdot \frac{x}{103.53}\\ 103.53\cdot cos75°=x\end{array}$

Simplify it with help of calculator,

  $\begin{array}{c}103.53\cdot cos75°=103.53\cdot \frac{x}{103.53}\\ 103.53\cdot cos75°=x\\ x=26.79\end{array}$

Therefore, the length of adjacent side is $26.79$ units.

Observe that the angle A and C are complementary angles so,

  $\begin{array}{c}\angle A+\angle C=90°\\ \angle A+75°=90°\\ \angle A=15°\end{array}$

Triangle so formed is provided below,

Hence, the sides of right angled triangle are $100,26.79\text{ and }103.53$ .