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

The given equation is \( 2x + 3\tan x = \pi \). Rearranging the terms yields \( \tan x = \frac{\pi - 2x}{3} \). Let \( f(x) = \tan x \) and \( g(x) = \frac{\pi - 2x}{3} \). We aim to determine the number of intersection points between these two functions within the interval \( [-2\pi, 2\pi] \). The function \( f(x) = \tan x \) has vertical asymptotes at \( x = \pm \frac{\pi}{2} \) and \( x = \pm \frac{3\pi}{2} \). The function \( g(x) = \frac{\pi - 2x}{3} \) is a linear function with a slope of \( -\frac{2}{3} \) and a y-intercept of \( \frac{\pi}{3} \). We can analyze the intersection points by graphical representation or by examining intervals. Within the interval \( [-2\pi, 2\pi] \), the intervals to consider are \( [-2\pi, -\frac{3\pi}{2}) \), \( (-\frac{3\pi}{2}, -\frac{\pi}{2}) \), \( (-\frac{\pi}{2}, \frac{\pi}{2}) \), \( (\frac{\pi}{2}, \frac{3\pi}{2}) \), and \( (\frac{3\pi}{2}, 2\pi] \). By sketching the graphs or analyzing the behavior of the functions in each interval, it is observed that there are 5 intersection points.
 
Consequently, the number of solutions is 5.