Chapter 6, Problem 11T

Express θ in each figure in terms of x.

1. (a)
2. (b)

To determine

To find: The angle $\theta$ in terms of $x$.

Answer to Problem 11T

The angle $\theta$ is ${\tan }^{-1}\frac{x}{4}$.

Explanation of Solution

Given:

A right angled triangle with opposite side as $x$ and the adjacent side as $4$.

Formula used:

The trigonometric functions of the angle $\theta$,

$\sin \theta =\frac{opposite}{hypotenuse}$, $\cos \theta =\frac{adjacent}{hypotenuse}$, $\tan \theta =\frac{opposite}{adjacent}$

Calculation:

The given sides are opposite side and an adjacent side.

Thus the given sides are for tangent function.

The tangent function is,

$\tan \theta =\frac{opposite}{\mathrm{adjacent}}$

Substitute $x$ for opposite and $4$ for adjacent.

$\begin{array}{c}\tan \theta =\frac{x}{4}\\ \theta ={\tan }^{-1}\frac{x}{4}\end{array}$

The angle $\theta$ in terms of $x$ is ${\tan }^{-1}\frac{x}{4}$.

To determine

To find: The angle $\theta$ in terms of $x$.

Answer to Problem 11T

The angle $\theta$ is ${\cos }^{-1}\frac{3}{x}$.

Explanation of Solution

Given:

A right angled triangle with adjacent side as $3$ and hypotenuse as $x$.

Formula used:

The trigonometric functions of the angle $\theta$,

$\sin \theta =\frac{opposite}{hypotenuse}$, $\cos \theta =\frac{adjacent}{hypotenuse}$, $\tan \theta =\frac{opposite}{adjacent}$

Calculation:

The given sides are adjacent side and hypotenuse.

Thus the given sides are for cosine function.

The cosine function is,

$\cos \theta =\frac{\mathrm{adjacent}}{hypotenuse}$

Substitute $3$ for adjacent and $x$ for hypotenuse

$\begin{array}{c}\cos \theta =\frac{3}{x}\\ \theta ={\cos }^{-1}\frac{3}{x}\end{array}$

The angle $\theta$ in terms of $x$ is ${\cos }^{-1}\frac{3}{x}$.