The number of solutions of the equation $2x+3\tan x=\pi$ in $x \in [-2\pi, 2\pi] – \left\{ \pm \frac{\pi}{2}, \pm \frac{3\pi}{2} \right\}$ is
- (1) 6
- (2) 5
- (3) 4
- (4) 3
Solution:
$$2x + 3\tan x = \pi$$
$$3\tan x = \pi – 2x$$
$$\tan x = \frac{\pi}{3} – \frac{2x}{3}$$
To find the number of solutions, we can find the number of intersection points of the curves $y = \tan x$ and the line $y = \frac{\pi}{3} – \frac{2x}{3}$ in the interval $[-2\pi, 2\pi]$.
The graph of $y = \tan x$ consists of separate branches divided by vertical asymptotes at $x = -\frac{3\pi}{2}, -\frac{\pi}{2}, \frac{\pi}{2}, \frac{3\pi}{2}$.
The line $y = -\frac{2}{3}x + \frac{\pi}{3}$ has a negative slope and goes downwards from left to right.
By plotting both graphs in the interval $[-2\pi, 2\pi]$, we can observe that the straight line intersects the tangent curve exactly once in each of its branches within the given domain.
The intersection points occur in the following intervals:
$$1^{\text{st}} \text{ intersection in } \left(-2\pi, -\frac{3\pi}{2}\right)$$
$$2^{\text{nd}} \text{ intersection in } \left(-\frac{3\pi}{2}, -\frac{\pi}{2}\right)$$
$$3^{\text{rd}} \text{ intersection in } \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$$
$$4^{\text{th}} \text{ intersection in } \left(\frac{\pi}{2}, \frac{3\pi}{2}\right)$$
$$5^{\text{th}} \text{ intersection in } \left(\frac{3\pi}{2}, 2\pi\right)$$
Total number of intersection points $= 5$
Ans. (2)