Find the number of solution of the equations
`x+ 2 tan x =(pi)/(2)`, when `x in[0,2pi]`

Answer

To find the number of solutions for the equation \( x + 2 \tan x = \frac{\pi}{2} \) in the interval \( [0, 2\pi] \), we can follow these steps:

### Step 1: Rearranging the Equation
We start with the equation:
\[ x + 2 \tan x = \frac{\pi}{2} \]
We can rearrange this to isolate \( \tan x \):
\[ 2 \tan x = \frac{\pi}{2} - x \]
Thus,
\[ \tan x = \frac{\frac{\pi}{2} - x}{2} \]

### Step 2: Defining Functions
Let’s define two functions based on our rearranged equation:
\[ y_1 = 2 \tan x \]
\[ y_2 = \frac{\pi}{2} - x \]
We need to find the points where these two functions intersect.

### Step 3: Analyzing the Functions
- The function \( y_2 = \frac{\pi}{2} - x \) is a straight line that decreases from \( \frac{\pi}{2} \) at \( x = 0 \) to \( -\frac{3\pi}{2} \) at \( x = 2\pi \).
- The function \( y_1 = 2 \tan x \) is periodic with vertical asymptotes at \( x = \frac{\pi}{2} + n\pi \) for \( n \in \mathbb{Z} \). Within the interval \( [0, 2\pi] \), it has vertical asymptotes at \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \).

### Step 4: Graphing the Functions
To visualize the solutions, we can sketch the graphs of \( y_1 \) and \( y_2 \):
- For \( y_2 \), plot a straight line from \( (0, \frac{\pi}{2}) \) to \( (2\pi, -\frac{3\pi}{2}) \).
- For \( y_1 \), plot the curve of \( 2 \tan x \), noting that it approaches \( +\infty \) as \( x \) approaches \( \frac{\pi}{2} \) from the left, and \( -\infty \) as \( x \) approaches \( \frac{\pi}{2} \) from the right. The same behavior occurs at \( x = \frac{3\pi}{2} \).

### Step 5: Finding Intersections
Now we analyze the intersections:
- In the interval \( [0, \frac{\pi}{2}) \), \( y_1 \) starts from 0 and goes to \( +\infty \), while \( y_2 \) starts from \( \frac{\pi}{2} \) and decreases. This indicates one intersection.
- In the interval \( (\frac{\pi}{2}, \frac{3\pi}{2}) \), \( y_1 \) starts from \( -\infty \) and goes to \( +\infty \), while \( y_2 \) continues to decrease. This indicates another intersection.
- In the interval \( (\frac{3\pi}{2}, 2\pi] \), \( y_1 \) starts from \( -\infty \) and goes to \( +\infty \), while \( y_2 \) continues to decrease. This indicates a third intersection.

### Conclusion
Thus, we find that there are three points of intersection, which means there are three solutions to the equation \( x + 2 \tan x = \frac{\pi}{2} \) in the interval \( [0, 2\pi] \).

### Final Answer
The number of solutions is \( 3 \).

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