Chapter 12.7, Problem 41HP

To determine

To find: Domain and range of $y=a\cos \theta$ and $y=a\sec \theta$ .

Answer to Problem 41HP

Domain of $y=a\cos \theta$ is set of all real numbers.

Range of the function $y=a\cos \theta$ is $\left[-a,a\right]$

Domain of the function $y=a\sec \theta$ is $R-\frac{k\pi }{2}$ .

Range of the function $y=a\sec \theta$ is $y\le -a$ and $y\ge a$ .

Explanation of Solution

Given information:

  $a$ is any positive real number.

  $y=a\cos \theta$

We know that the domain of a function is the set of all possible inputs for the function.

Therefore,

For all value of $\theta$ above function exists.

Therefore,

Domain of $y=a\cos \theta$ is set of all real numbers.

Range of $y=a\cos \theta$ is

We know that range of $\cos \theta$ is $\left[-1,1\right]$

  $\Rightarrow -1\le \cos \theta \le 1$

Multiplying with $a$ on both sides

We get

  $\begin{array}{l}\Rightarrow -1\times a\le a\times \cos \theta \le 1\times a\\ \Rightarrow -a\le a\cos \theta \le a\end{array}$

Therefore,

Range of the function $y=a\cos \theta$ is $\left[-a,a\right]$

  $y=a\sec \theta$

We know that the domain of a function is the set of all possible inputs for the function.

Therefore,

For all value of $\theta$ above function exists except the values where $\cos \theta =0$ because at $\cos \theta =0$ function does not exists.

Therefore,

Domain of the function $y=a\sec \theta$ is $R-\frac{k\pi }{2}$ .

Where $k$ is an odd integer.

We know that range of secant function is

  $\sec \theta \ge 1$ or $\sec \theta \le -1$

Multiplying with $a$ on both sides

We get

  $\Rightarrow$ $a\sec \theta \ge a$ or $a\sec \theta \le -a$

Therefore,

Range of the function $y=a\sec \theta$ is $y\le -a$ and $y\ge a$ .