Find the principal solutions of each of the following equations :
(i) `sinx=(sqrt(3))/(2)`
(ii) `cosx=(1)/(2)`
(iii) `tanx=sqrt(3)`
(iv) `cotx=sqrt(3)`
(v) `"cosec "x=sqrt(3)`
(vi) `secx=(2)/(sqrt(3))`

Answer

To find the principal solutions for each of the given equations, we will use the properties of trigonometric functions and their values in different quadrants. The principal solutions are the angles \( x \) in the range \( [0, 2\pi) \).

### Step-by-Step Solutions:

#### (i) \( \sin x = \frac{\sqrt{3}}{2} \)

1. **Identify the quadrants:** The sine function is positive in the first and second quadrants.
2. **Find the reference angle:** The reference angle for \( \sin x = \frac{\sqrt{3}}{2} \) is \( \frac{\pi}{3} \).
3. **Determine the solutions:**
- In the first quadrant: \( x = \frac{\pi}{3} \)
- In the second quadrant: \( x = \pi - \frac{\pi}{3} = \frac{2\pi}{3} \)

**Principal Solutions:** \( x = \frac{\pi}{3}, \frac{2\pi}{3} \)

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#### (ii) \( \cos x = \frac{1}{2} \)

1. **Identify the quadrants:** The cosine function is positive in the first and fourth quadrants.
2. **Find the reference angle:** The reference angle for \( \cos x = \frac{1}{2} \) is \( \frac{\pi}{3} \).
3. **Determine the solutions:**
- In the first quadrant: \( x = \frac{\pi}{3} \)
- In the fourth quadrant: \( x = 2\pi - \frac{\pi}{3} = \frac{5\pi}{3} \)

**Principal Solutions:** \( x = \frac{\pi}{3}, \frac{5\pi}{3} \)

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#### (iii) \( \tan x = \sqrt{3} \)

1. **Identify the quadrants:** The tangent function is positive in the first and third quadrants.
2. **Find the reference angle:** The reference angle for \( \tan x = \sqrt{3} \) is \( \frac{\pi}{3} \).
3. **Determine the solutions:**
- In the first quadrant: \( x = \frac{\pi}{3} \)
- In the third quadrant: \( x = \pi + \frac{\pi}{3} = \frac{4\pi}{3} \)

**Principal Solutions:** \( x = \frac{\pi}{3}, \frac{4\pi}{3} \)

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#### (iv) \( \cot x = \sqrt{3} \)

1. **Identify the quadrants:** The cotangent function is positive in the first and third quadrants.
2. **Find the reference angle:** The reference angle for \( \cot x = \sqrt{3} \) is \( \frac{\pi}{6} \).
3. **Determine the solutions:**
- In the first quadrant: \( x = \frac{\pi}{6} \)
- In the third quadrant: \( x = \pi + \frac{\pi}{6} = \frac{7\pi}{6} \)

**Principal Solutions:** \( x = \frac{\pi}{6}, \frac{7\pi}{6} \)

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#### (v) \( \csc x = \sqrt{3} \)

1. **Convert to sine:** Since \( \csc x = \frac{1}{\sin x} \), we have \( \sin x = \frac{1}{\sqrt{3}} \).
2. **Identify the quadrants:** The sine function is positive in the first and second quadrants.
3. **Find the reference angle:** The reference angle for \( \sin x = \frac{1}{\sqrt{3}} \) is \( \frac{\pi}{6} \).
4. **Determine the solutions:**
- In the first quadrant: \( x = \frac{\pi}{6} \)
- In the second quadrant: \( x = \pi - \frac{\pi}{6} = \frac{5\pi}{6} \)

**Principal Solutions:** \( x = \frac{\pi}{6}, \frac{5\pi}{6} \)

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#### (vi) \( \sec x = \frac{2}{\sqrt{3}} \)

1. **Convert to cosine:** Since \( \sec x = \frac{1}{\cos x} \), we have \( \cos x = \frac{\sqrt{3}}{2} \).
2. **Identify the quadrants:** The cosine function is positive in the first and fourth quadrants.
3. **Find the reference angle:** The reference angle for \( \cos x = \frac{\sqrt{3}}{2} \) is \( \frac{\pi}{6} \).
4. **Determine the solutions:**
- In the first quadrant: \( x = \frac{\pi}{6} \)
- In the fourth quadrant: \( x = 2\pi - \frac{\pi}{6} = \frac{11\pi}{6} \)

**Principal Solutions:** \( x = \frac{\pi}{6}, \frac{11\pi}{6} \)

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