Find the principal values of the following :
(i) `sin^(-1)(sin.(5pi)/(3))` (ii) `cos^(-1)cos((4pi)/(3))` (iii) ` cos [(pi)/(3) + cos^(-1) (-(1)/(2))]`

Answer

To solve the given problems step by step, we will find the principal values of the following expressions:

### (i) \( \sin^{-1}(\sin(5\pi/3)) \)

1. **Identify the angle**: The angle \( 5\pi/3 \) is in the fourth quadrant. To find its equivalent angle in the range of \( -\pi/2 \) to \( \pi/2 \), we can subtract \( 2\pi \) (or \( 360^\circ \)) from it.
\[ 5\pi/3 - 2\pi = 5\pi/3 - 6\pi/3 = -\pi/3 \]

2. **Apply the sine function**:
\[ \sin(5\pi/3) = \sin(-\pi/3) \]

3. **Use the property of inverse sine**:
\[ \sin^{-1}(\sin(5\pi/3)) = \sin^{-1}(\sin(-\pi/3)) = -\pi/3 \]

**Final Answer**: \( \sin^{-1}(\sin(5\pi/3)) = -\pi/3 \)

### (ii) \( \cos^{-1}(\cos(4\pi/3)) \)

1. **Identify the angle**: The angle \( 4\pi/3 \) is in the third quadrant. To find its equivalent angle in the range of \( 0 \) to \( \pi \), we can use the property of cosine:
\[ \cos(4\pi/3) = \cos(\pi + \pi/3) = -\cos(\pi/3) \]

2. **Use the property of inverse cosine**:
\[ \cos^{-1}(\cos(4\pi/3)) = \cos^{-1}(-\cos(\pi/3)) \]

3. **Find the angle**: Since \( \cos^{-1}(-\cos(\pi/3)) \) corresponds to \( \pi - \pi/3 \):
\[ \pi - \pi/3 = 2\pi/3 \]

**Final Answer**: \( \cos^{-1}(\cos(4\pi/3)) = 2\pi/3 \)

### (iii) \( \cos\left(\frac{\pi}{3} + \cos^{-1}\left(-\frac{1}{2}\right)\right) \)

1. **Find \( \cos^{-1}(-1/2) \)**: The angle whose cosine is \( -1/2 \) in the range \( [0, \pi] \) is \( 2\pi/3 \).
\[ \cos^{-1}(-1/2) = 2\pi/3 \]

2. **Substitute back into the expression**:
\[ \cos\left(\frac{\pi}{3} + \frac{2\pi}{3}\right) = \cos(\pi) \]

3. **Evaluate \( \cos(\pi) \)**:
\[ \cos(\pi) = -1 \]

**Final Answer**: \( \cos\left(\frac{\pi}{3} + \cos^{-1}\left(-\frac{1}{2}\right)\right) = -1 \)

### Summary of Answers:
1. \( \sin^{-1}(\sin(5\pi/3)) = -\pi/3 \)
2. \( \cos^{-1}(\cos(4\pi/3)) = 2\pi/3 \)
3. \( \cos\left(\frac{\pi}{3} + \cos^{-1}\left(-\frac{1}{2}\right)\right) = -1 \)