Chapter 6.2, Problem 18E

To determine

To calculate: The value of x and y in terms of trigonometric ratios provided in the triangle.

Answer to Problem 18E

The value in terms of trigonometric ratios are $4\tan \theta =x$ and $y=\frac{4}{\cos \theta }$ .

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 opposite side is of length x units, adjacent side is of length 4 units and hypotenuse has length y units.

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 tangent function is,

  $\begin{array}{c}\tan \theta =\frac{\text{opposite}}{\text{adjacent}}\\ \tan \theta =\frac{x}{4}\end{array}$

Multiply both sides by 4,

  $\begin{array}{c}4\tan \theta =4\cdot \frac{x}{4}\\ 4\tan \theta =x\end{array}$

The value of cosine function is,

  $\begin{array}{c}cos\theta =\frac{\text{adjacent}}{\text{hypotenuse}}\\ cos\theta =\frac{4}{y}\end{array}$

Multiply both sides by y ,

  $\begin{array}{c}y\cos \theta =y\times \frac{4}{y}\\ y\cos \theta =4\end{array}$

Divide both sides by $cos\theta$

  $y=\frac{4}{\cos \theta }$

Hence, the value in terms of trigonometric ratios are $4\tan \theta =x$ and $y=\frac{4}{\cos \theta }$ .