Find the principal values of the following :
(i) `sin^(-1)(sqrt(3))`
(ii) `cot^(-1)(-sqrt(3))`
(iii) `cos^(-1)(-1(1)/(2))`
(iv) `sec^(-1)(-(2)/sqrt(3))`
(v) `tan^(-1)(-1)`
(vi) `cosec ^(-1)(-1)`.

Answer

To find the principal values of the given inverse trigonometric functions, we will follow the standard ranges for each function. Let's solve each part step by step.

### (i) Find `sin^(-1)(sqrt(3))`

1. **Identify the value**: We need to find the angle whose sine is `sqrt(3)/2`.
2. **Determine the angle**: The angle that satisfies this condition is `π/3` (or 60 degrees).
3. **Principal value**: Since `sqrt(3)` is positive, the principal value is:
\[ \sin^{-1}(\sqrt{3}/2) = \frac{\pi}{3} \]

### (ii) Find `cot^(-1)(-sqrt(3))`

1. **Identify the value**: We need to find the angle whose cotangent is `-sqrt(3)`.
2. **Determine the angle**: The angle that satisfies this condition is `5π/6` (or 150 degrees), since cotangent is negative in the second quadrant.
3. **Principal value**: Thus, the principal value is:
\[ \cot^{-1}(-\sqrt{3}) = \frac{5\pi}{6} \]

### (iii) Find `cos^(-1)(-1/2)`

1. **Identify the value**: We need to find the angle whose cosine is `-1/2`.
2. **Determine the angle**: The angle that satisfies this condition is `2π/3` (or 120 degrees), since cosine is negative in the second quadrant.
3. **Principal value**: Therefore, the principal value is:
\[ \cos^{-1}(-\frac{1}{2}) = \frac{2\pi}{3} \]

### (iv) Find `sec^(-1)(-2/sqrt(3))`

1. **Identify the value**: We need to find the angle whose secant is `-2/sqrt(3)`.
2. **Determine the angle**: The corresponding angle is `5π/6` (or 150 degrees), since secant is negative in the second quadrant.
3. **Principal value**: Thus, the principal value is:
\[ \sec^{-1}\left(-\frac{2}{\sqrt{3}}\right) = \frac{5\pi}{6} \]

### (v) Find `tan^(-1)(-1)`

1. **Identify the value**: We need to find the angle whose tangent is `-1`.
2. **Determine the angle**: The angle that satisfies this condition is `-π/4` (or -45 degrees), but we need the principal value in the range of `(-π/2, π/2)`, which is `3π/4`.
3. **Principal value**: Therefore, the principal value is:
\[ \tan^{-1}(-1) = -\frac{\pi}{4} \]

### (vi) Find `cosec^(-1)(-1)`

1. **Identify the value**: We need to find the angle whose cosecant is `-1`.
2. **Determine the angle**: The angle that satisfies this condition is `-π/2` (or -90 degrees), but we need the principal value in the range of `(-π/2, π/2)`.
3. **Principal value**: Thus, the principal value is:
\[ \csc^{-1}(-1) = -\frac{\pi}{2} \]

### Summary of Principal Values
1. \( \sin^{-1}(\sqrt{3}/2) = \frac{\pi}{3} \)
2. \( \cot^{-1}(-\sqrt{3}) = \frac{5\pi}{6} \)
3. \( \cos^{-1}(-1/2) = \frac{2\pi}{3} \)
4. \( \sec^{-1}(-2/\sqrt{3}) = \frac{5\pi}{6} \)
5. \( \tan^{-1}(-1) = -\frac{\pi}{4} \)
6. \( \csc^{-1}(-1) = -\frac{\pi}{2} \)