Q. 99 - Precalculus Enhanced with Graphing Utilities

Question

The graphs of the given pairs of functions intersect infinitely many times. Find four of these points of intersection.  

$$\begin{gathered} y=\tan x \\ y=1 \end{gathered}$$

Step-by-Step Solution

Answer

The four points of intersection are $\left(\frac{\mathrm{\pi }}{4},1\right),\left(\frac{5\mathrm{\pi }}{4},1\right),\left(\frac{9\mathrm{\pi }}{4},1\right),\left(\frac{13\mathrm{\pi }}{4},1\right)$.

Step 1. Given Information

We are given two equations $y=\tan x$ and $y=1$.

We need to find any four points of intersection of the two graphs.

We will graph the two equations and then find the first point of intersection and then using the properties of tangent function we will find the other three points.

Step 2. Graph the functions

The graph of the two equations is given as 

It can be seen that the line and the curve intersect each other at infinitely many points and the first point of intersection is $\left(\frac{\mathrm{\pi }}{4},1\right)$.

Step 3. Find the other points

The tangent function is a cyclic function. So

$$\begin{gathered} \tan x=\tan (\mathrm{\pi }+\mathrm{x}) \\ \mathrm{tanx}=\tan (2\mathrm{\pi }+\mathrm{x}) \\ \mathrm{tanx}=\tan (3\mathrm{\pi }+\mathrm{x}) \end{gathered}$$

So the x coordinates of the other points are

$\mathrm{\pi }+\frac{\mathrm{\pi }}{4}=\frac{5\mathrm{\pi }}{4}$ or

$2\mathrm{\pi }+\frac{\mathrm{\pi }}{4}=\frac{9\mathrm{\pi }}{4}$ or

$3\mathrm{\pi }+\frac{\mathrm{\pi }}{4}=\frac{13\mathrm{\pi }}{4}$

Thus the four intersecting points are $\left(\frac{\mathrm{\pi }}{4},1\right),\left(\frac{5\mathrm{\pi }}{4},1\right),\left(\frac{9\mathrm{\pi }}{4},1\right),\left(\frac{13\mathrm{\pi }}{4},1\right)$.