Animated Solution for Mathematics - Trigonometry: The number of solutions of the equation $2x+3\tan x=\pi ,x\in [-2\pi ,2\pi ]-\{\pm \frac{\pi }{2},\pm \frac{3\pi }{2}\}$ is

The equation $2x+3\tan x=\pi$ is a transcendental equation, meaning it cannot be solved using standard algebraic manipulation. To find the number of solutions within the domain $x\in [-2\pi ,2\pi ]$, we must employ graphical analysis.

By rearranging the equation to isolate the trigonometric component, we obtain:

Dividing by $3$, we arrive at the following form:

We define two functions: $f(x)=\tan x$ and $g(x)=\frac{\pi }{3}-\frac{2x}{3}$. The number of solutions to the original equation is equivalent to the number of intersection points between these two graphs.

The function $f(x)=\tan x$ is undefined at odd multiples of $\frac{\pi }{2}$. Within the domain $x\in [-2\pi ,2\pi ]$, these vertical asymptotes occur at:

These four vertical lines act as barriers, partitioning the domain into five distinct, continuous intervals. Within each interval, the graph of $\tan x$ is a strictly increasing curve that spans from $-\infty$ to $+\infty$.

The second function, $g(x)=\frac{\pi }{3}-\frac{2x}{3}$, is a linear function with a constant negative slope of $-\frac{2}{3}$. Because $g(x)$ is strictly decreasing while $f(x)=\tan x$ is strictly increasing on each of the five intervals, they must intersect.

Specifically, in each of the five intervals, the curve $f(x)$ starts at $-\infty$ and climbs to $+\infty$, while the line $g(x)$ descends through the interval. By the Intermediate Value Theorem, the functions are guaranteed to cross exactly once in each interval.

Since there are five distinct intervals created by the asymptotes, and each interval contains exactly one intersection point, we can conclude the total count.

The number of intersection points is $5$. Therefore, the total number of solutions to the equation $2x+3\tan x=\pi$ in the domain $x\in [-2\pi ,2\pi ]$ is 5.