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

A Geometric Odyssey

Welcome, fellow traveler of the JEE landscape. Today, we face a problem that initially looks like a simple algebraic equation: $2x+3\tan x=\pi$.

But look closer. Do you see the trap? We have $x$ inside the tangent function and $x$ outside as a linear term.

This is a transcendental equation. You cannot isolate $x$ using standard algebraic manipulation. If you try, you will find yourself running in circles. But fear not—this is where we pivot from algebra to the elegance of geometry.

The Separation of Powers

When algebra fails, geometry prevails. Our first move is to isolate the functions. We want to see how two distinct curves interact. Let us rearrange our equation:

Dividing by $3$, we get:

Now, we define our two combatants. Let $f(x)=\tan x$ be our trigonometric curve, and let $g(x)=\frac{\pi }{3}-\frac{2x}{3}$ be our linear function.

The number of solutions to the original equation is exactly the number of times these two graphs intersect. We have turned a terrifying equation into a visual puzzle.

Mapping the Territory

Before we draw, we must understand the boundaries. Our domain is $x\in [-2\pi ,2\pi ]$, but we must exclude the points where $\tan x$ explodes to infinity: $x=\pm \frac{\pi }{2}$ and $x=\pm \frac{3\pi }{2}$.

These are our vertical asymptotes. Imagine them as impenetrable walls. They divide our domain into five distinct 'rooms' or sub-intervals:

1. $[-2\pi ,-\frac{3\pi }{2})$ 2. $(-\frac{3\pi }{2},-\frac{\pi }{2})$ 3. $(-\frac{\pi }{2},\frac{\pi }{2})$ 4. $(\frac{\pi }{2},\frac{3\pi }{2})$ 5. $(\frac{3\pi }{2},2\pi ]$

The Collision Course

Now, let us analyze the linear function $g(x)=\frac{\pi }{3}-\frac{2x}{3}$. It has a negative slope, meaning it is constantly falling as we move from left to right.

At $x=0$, it sits at $y=\frac{\pi }{3}\approx 1.05$. At the right boundary $x=2\pi$, it drops to $y=-\pi \approx -3.14$. This line is a steady, downward-sloping traveler.

Now, consider the tangent branches. In every single one of those five intervals, the tangent function sweeps from $-\infty$ to $+\infty$.

Because our line $g(x)$ is continuous and finite, it acts like a bridge crossing the infinite chasm of the tangent curve. In every single interval, the tangent curve must cross our line exactly once. It is mathematically impossible for them to miss each other.

The Final Tally

Let us count them carefully:

- In the first interval $[-2\pi ,-\frac{3\pi }{2})$, the tangent curve rises from $0$ to $\infty$, and our line is positive. They intersect once.

- In the second, third, and fourth intervals, the tangent curve sweeps the entire vertical range $(-\infty ,\infty )$. Our line, being finite, must cross this sweep exactly once in each interval. That gives us three more intersections.

- Finally, in the fifth interval $(\frac{3\pi }{2},2\pi ]$, the tangent curve comes up from $-\infty$ to $0$. Our line is below the x-axis at this point. They collide one last time.

One, two, three, four, five. Five intersections. Five solutions.

We have conquered the transcendental equation not by brute force, but by visualizing the dance between a line and a curve. Remember this: whenever you see a mix of trig and algebra, stop calculating and start sketching. The geometry will always show you the way.