Animated Solution for Mathematics - Trigonometry: The number of solutions of the equation $x+2\tan x=\frac{\pi }{2}$ in the interval $[0,2\pi ]$ is :

We are tasked with finding the number of solutions to the equation $x+2\tan x=\frac{\pi }{2}$ within the interval $[0,2\pi ]$.

At first glance, you might be tempted to reach for your algebraic toolkit, trying to isolate $x$. But stop right there! This is a transcendental equation, and trying to solve it using standard algebra is like trying to cut a diamond with a butter knife.

It is simply not the right tool for the job. Instead, we are going to use the most powerful weapon in our arsenal: the graphical method.

To use the graphical method effectively, we must first untangle the knot. We want to separate the trigonometric part from the algebraic part.

Let us take our original equation, $x+2\tan x=\frac{\pi }{2}$, and rearrange it. By moving $x$ to the right side, we get $2\tan x=\frac{\pi }{2}-x$.

Now, let us divide the entire equation by $2$. This gives us a much cleaner form:

By doing this, we have transformed a single, intimidating equation into a beautiful intersection problem. We are now looking for the points where the function $y_1=\tan x$ meets the line $y_2=-\frac{1}{2}x+\frac{\pi }{4}$.

Let us start by plotting our first function, $y_1=\tan x$. Before we draw the curve, we must respect the boundaries.

The tangent function is notorious for its vertical asymptotes, which occur wherever $\cos x=0$. In our interval $[0,2\pi ]$, these occur at $x=\frac{\pi }{2}$ and $x=\frac{3\pi }{2}$. Let us mark these with dashed lines.

Now, let us trace the branches of $\tan x$: In the first quadrant, from $0$ to $\frac{\pi }{2}$, the curve shoots upwards from $0$ to positive infinity. In the second and third quadrants, from $\frac{\pi }{2}$ to $\frac{3\pi }{2}$, it emerges from negative infinity, crosses the x-axis at $\pi$, and climbs to positive infinity. * Finally, from $\frac{3\pi }{2}$ to $2\pi$, it rises from negative infinity and ends at $0$.

Now, consider our second function, $y_2=-\frac{1}{2}x+\frac{\pi }{4}$. This is a straight line with a negative slope of $-\frac{1}{2}$, meaning it is strictly decreasing.

At $x=0$, the y-intercept is $\frac{\pi }{4}$. If we set $y=0$, we find the x-intercept is at $x=\frac{\pi }{2}$. This line is our steady, predictable guide through the wild landscape of the tangent function.

Now, let us count the intersections:

1. In the first interval, $[0,\frac{\pi }{2})$: The tangent curve is strictly increasing from $0$ to $\infty$, while our line is decreasing from $\frac{\pi }{4}$ to $0$. They are destined to cross exactly once. That is our first solution!

2. In the middle section, $(\frac{\pi }{2},\frac{3\pi }{2})$: The tangent curve sweeps from $-\infty$ to $\infty$, covering every possible real value. Our line is just quietly continuing its downward path. Because the tangent curve covers all real numbers here, it must intersect our line exactly once. That is our second solution!

3. In the last interval, $(\frac{3\pi }{2},2\pi ]$: The tangent curve rises from $-\infty$ to $0$. Our line is already in negative territory and continues to decrease. Again, the continuous nature of these functions guarantees one final intersection. That is our third solution!

By systematically analyzing these intervals, we have found exactly three points of intersection. The graphical approach has turned a complex problem into a clear, visual victory.

The final answer is 3.