Chapter 6.2, Problem 35E

To determine

To calculate: The sides of the right angled triangle.

Answer to Problem 35E

The triangle along with sides and acute angles is,

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 hypotenuse is 1000 units.

Label the length of opposite side of the triangle as x and adjacent side 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 cosine function is,

  $\begin{array}{c}cos\frac{\pi }{8}=\frac{\text{adjacent}}{\text{hypotenuse}}\\ cos\frac{\pi }{8}=\frac{y}{33.5}\end{array}$

Divide both sides by $35.5$ ,

  $\begin{array}{c}33.5cos\frac{\pi }{8}=33.5\cdot \frac{y}{33.5}\\ 33.5cos\frac{\pi }{8}=y\end{array}$

Simplify it with help of calculator,

  $\begin{array}{c}33.5cos\frac{\pi }{8}=35.5\cdot \frac{y}{35.5}\\ 33.5cos\frac{\pi }{8}=y\\ y=30.95\end{array}$

Therefore, the length of adjacent side is $30.95$ units.

The value of sine function is,

  $\begin{array}{c}\sin \frac{\pi }{8}=\frac{\text{opposite}}{\text{hypotenuse}}\\ \sin \frac{\pi }{8}=\frac{x}{33.5}\end{array}$

Multiply both sides by $33.5$ ,

  $\begin{array}{c}33.5\sin \frac{\pi }{8}=33.5\cdot \frac{x}{33.5}\\ 33.5\sin \frac{\pi }{8}=x\end{array}$

Simplify it with help of calculator,

  $\begin{array}{c}33.5\sin \frac{\pi }{8}=33.5\cdot \frac{x}{33.5}\\ 33.5\sin \frac{\pi }{8}=x\\ x=12.82\end{array}$

Therefore, the length of opposite side is $12.82$ units.

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

  $\begin{array}{c}\angle A+\angle C=90°\\ \angle A+\frac{\pi }{8}=90°\\ \angle A=\frac{3\pi }{8}\end{array}$

Thus, the triangle so formed is provided below,