Question

The number of solutions of the equation $ 2x + 3\tan x = \pi $, $ x \in [-2\pi, 2\pi] - \left\{ \pm \frac{\pi}{2}, \pm \frac{3\pi}{2} \right\} $ is

• A. 6
• B. 5
• C. 4
• D. 3

Hint:

To find the number of solutions of an equation involving trigonometric and linear functions, sketch the graphs of both functions and count the number of intersection points.

Answer 1

The Correct Option is B

Given equation is \( 2x + 3\tan x = \pi \)
Rearranging the terms, we get \( \tan x = \frac{\pi - 2x}{3} \) 
Let \( f(x) = \tan x \) and \( g(x) = \frac{\pi - 2x}{3} \) 
We need to find the number of intersection points of these two functions in the given interval \( [-2\pi, 2\pi] \). 
\( f(x) = \tan x \) has vertical asymptotes at \( x = \pm \frac{\pi}{2} \) and \( x = \pm \frac{3\pi}{2} \). \( g(x) = \frac{\pi - 2x}{3} \) is a straight line with slope \( -\frac{2}{3} \) and y-intercept \( \frac{\pi}{3} \). 
We can analyze the intersection points graphically or by analyzing intervals. In the interval \( [-2\pi, 2\pi] \), we have the following intervals to consider: \( [-2\pi, -\frac{3\pi}{2}) \), \( (-\frac{3\pi}{2}, -\frac{\pi}{2}) \), \( (-\frac{\pi}{2}, \frac{\pi}{2}) \), \( (\frac{\pi}{2}, \frac{3\pi}{2}) \), \( (\frac{3\pi}{2}, 2\pi] \). 
By sketching the graphs or analyzing the behavior of the functions in each interval, we can observe that there are 5 intersection points.
  
Therefore, the number of solutions is 5.

Answer 2

To find the number of solutions of the equation \(2x + 3\tan x = \pi\) in the interval \(x \in [-2\pi, 2\pi] - \left\{ \pm \frac{\pi}{2}, \pm \frac{3\pi}{2} \right\}\), we need to solve the equation for each region defined by the exclusion of these points where \( \tan x \) is undefined.

The given interval is divided by the points where \(\tan x\) is undefined: \(\pm \frac{\pi}{2}\) and \(\pm \frac{3\pi}{2}\). These points exclude certain regions. We analyze the function within these regions.

1. Since \(\tan x\) is periodic with period \(\pi\), we examine the behavior and number of solutions within intervals separated by \(\frac{\pi}{2}\).
2. In any interval \((n\pi - \frac{\pi}{2}, n\pi + \frac{\pi}{2})\), \(\tan x\) goes from \(-\infty\) to \(+\infty\).
3. The expression \(2x + 3\tan x = \pi\) can be graphed as a combination of a linear function \(2x\) and a cyclic function \(3\tan x\).
4. We'll find intersections between \(2x + 3\tan x\) and \(\pi\) graphically and then count the solutions over specified regions.

Let’s analyze each specific region:

• For the region \((-2\pi, -\frac{3\pi}{2})\), \((-2\pi, 2\pi)\) applies, and the tangent curve crosses \(\pi\) once.
• Repeat this analysis across \((-\frac{3\pi}{2}, -\frac{\pi}{2})\), \((-\frac{\pi}{2}, \frac{\pi}{2})\), \((\frac{\pi}{2}, \frac{3\pi}{2})\), and \((\frac{3\pi}{2}, 2\pi)\), identifying the behavior of the graphs, intersections apply at each region.

By analyzing these regions, it's evident that there are a total of 5 solutions where the function \(2x + 3\tan x = \pi\) crosses the horizontal line at \(\pi\) within the range \(x \in [-2\pi, 2\pi]\).

Thus, the correct answer is 5.