Chapter 13.8, Problem 53PPSE

Solution

i.

To graph: The given functions on the same axis.

Given:

The functions $y=\tan x$ and $y=\cot x$ .

Calculation:

The graph of $y=\tan x$ and $y=\cot x$ is given below:

  

Here the red line indicates the $y=\tan x$ and the blue line indicates $y=\cot x$ .

Conclusion:

Hence, the graph of functions is given above.

ii.

To find: The domain range and asymptote of the function.

The domain, range, and asymptote of $y=\tan x$ is $\frac{\pi }{2}+\pi n$ , $\left(-\infty ,\infty \right)$ , and $x=\frac{\pi }{2}+\pi n$ . And the domain range and asymptote of $y=\cot x$ is $\pi n$ , $\left(-\infty ,\infty \right)$ and $x=\pi n$ .

Given:

The functions $y=\tan x$ and $y=\cot x$ .

Concept Used:

1. The range represents the values of $y-\text{axis}$ which is covered by the graph of the function.
2. To find the asymptotes, obtain the value for which the function is undefined.

Calculation:

The domain of $y=\tan x$ is all real numbers except $\frac{\pi }{2}+\pi n$ because at this point the $\cos x$ is equivalent to zero. Here $n$ is a integer. Similarly, the domain of $y=\cot x$ is all real numbers except $\pi n$ because at this point the $\sin x$ is equivalent to zero.

The range of $y=\tan x$ is all real number because the curve can go $-\infty$ to infinity that can be seen in the graph. Similarly, the range of $y=\cot x$ is also for all real numbers according to the graph.

To find the vertical asymptote, obtain the value for which the function is undefined. It is understood that $y=\frac{\sin x}{\cos x}$ .The $\cos x$ is zero for $\frac{\pi }{2}+\pi n$ . It means at this point the function is undefined.

So, the vertical asymptote is $x=\frac{\pi }{2}+\pi n$ for integer n.

The horizontal and oblique asymptote does not exist for $y=\tan x$ .

Similarly, to find the vertical asymptote, obtain the value for which the function is undefined. It is understood that $y=\frac{\cot x}{\sin x}$ .The $\sin x$ is zero for $\pi n$ . It means at this point the function is undefined. So, the vertical asymptote of the function is $x=\pi n$ .

Conclusion:

The domain, range, and asymptote of $y=\tan x$ is $\frac{\pi }{2}+\pi n$ , $\left(-\infty ,\infty \right)$ , and $x=\frac{\pi }{2}+\pi n$ . And the domain range and asymptote of $y=\cot x$ is $\pi n$ , $\left(-\infty ,\infty \right)$ and $x=\pi n$ .

iii.

To find: The name for the reflection lines.

The name of the reflection lines are $\frac{\pi }{4}$ and $\frac{3\pi }{4}$ .

Given:

The functions $y=\tan x$ and $y=\cot x$ .

Concept Used:

1. The reflection line can be found by finding the intersection point of the function.

Calculation:

Consider the following graph:

  

The curves intersect point $\left(\frac{\pi }{4},1\right)$ and $\left(\frac{3\pi }{4},-1\right)$ so the name of the reflection lines are $\frac{\pi }{4}$ and $\frac{3\pi }{4}$ .

Conclusion:

The name of the reflection lines are $\frac{\pi }{4}$ and $\frac{3\pi }{4}$ .