Find the principal solution of the following equations :
(1) `sin x =(1)/(sqrt(2))`
(2) `tan=-1`
(3) `sqrt(3) cosec x +2=0`.

Answer

To find the principal solutions of the given equations, we will solve each equation step by step.

### Part (1): Solve `sin x = 1/sqrt(2)`

1. **Identify the angle whose sine is `1/sqrt(2)`.**
We know that `sin(π/4) = 1/sqrt(2)`.

2. **Determine the quadrants where sine is positive.**
Sine is positive in the first and second quadrants.

3. **Find the angles in those quadrants.**
- First quadrant: \( x = \frac{\pi}{4} \)
- Second quadrant: \( x = \pi - \frac{\pi}{4} = \frac{3\pi}{4} \)

4. **Write the principal solutions.**
The principal solutions for `sin x = 1/sqrt(2)` are:
\( x = \frac{\pi}{4}, \frac{3\pi}{4} \)

### Part (2): Solve `tan x = -1`

1. **Identify the angle whose tangent is `-1`.**
We know that `tan(π/4) = 1`, so `tan(π/4 + π) = -1`.

2. **Determine the quadrants where tangent is negative.**
Tangent is negative in the second and fourth quadrants.

3. **Find the angles in those quadrants.**
- Second quadrant: \( x = \pi - \frac{\pi}{4} = \frac{3\pi}{4} \)
- Fourth quadrant: \( x = 2\pi - \frac{\pi}{4} = \frac{7\pi}{4} \)

4. **Write the principal solutions.**
The principal solutions for `tan x = -1` are:
\( x = \frac{3\pi}{4}, \frac{7\pi}{4} \)

### Part (3): Solve `sqrt(3) cosec x + 2 = 0`

1. **Rearrange the equation.**
Start by isolating `cosec x`:
\( \sqrt{3} \cdot \csc x = -2 \)
\( \csc x = -\frac{2}{\sqrt{3}} \)

2. **Convert cosecant to sine.**
\( \csc x = \frac{1}{\sin x} \)
Thus, \( \sin x = -\frac{\sqrt{3}}{2} \)

3. **Determine the quadrants where sine is negative.**
Sine is negative in the third and fourth quadrants.

4. **Find the angles in those quadrants.**
- Reference angle: \( x = \frac{\pi}{3} \) (since \( \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2} \))
- Third quadrant: \( x = \pi + \frac{\pi}{3} = \frac{4\pi}{3} \)
- Fourth quadrant: \( x = 2\pi - \frac{\pi}{3} = \frac{5\pi}{3} \)

5. **Write the principal solutions.**
The principal solutions for `sqrt(3) cosec x + 2 = 0` are:
\( x = \frac{4\pi}{3}, \frac{5\pi}{3} \)

### Summary of Principal Solutions:
1. For `sin x = 1/sqrt(2)`: \( x = \frac{\pi}{4}, \frac{3\pi}{4} \)
2. For `tan x = -1`: \( x = \frac{3\pi}{4}, \frac{7\pi}{4} \)
3. For `sqrt(3) cosec x + 2 = 0`: \( x = \frac{4\pi}{3}, \frac{5\pi}{3} \)