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

Let's solve the given problem step by step.

### Question:
Find the principal values of the following:
1. \( \sin^{-1}(\sin(5\pi/3)) \)
2. \( \cos^{-1}(\cos(4\pi/3)) \)
3. \( \cos\left(\frac{\pi}{3} + \cos^{-1}\left(-\frac{1}{2}\right)\right) \)

### Step-by-Step Solution:

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

1. **Identify the angle**: We start with \( 5\pi/3 \). This angle is greater than \( 2\pi \), so we can find its equivalent angle within the range of \( [0, 2\pi] \).
\[ 5\pi/3 - 2\pi = 5\pi/3 - 6\pi/3 = -\pi/3 \]
Thus, \( \sin(5\pi/3) = \sin(-\pi/3) \).

2. **Calculate the sine value**:
\[ \sin(-\pi/3) = -\sin(\pi/3) = -\frac{\sqrt{3}}{2} \]

3. **Find the principal value**:
\[ \sin^{-1}(-\frac{\sqrt{3}}{2}) = -\frac{\pi}{3} \]
Therefore, the principal value is:
\[ \sin^{-1}(\sin(5\pi/3)) = -\frac{\pi}{3} \]

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

1. **Identify the angle**: The angle \( 4\pi/3 \) is in the third quadrant. We can express it as:
\[ 4\pi/3 = \pi + \pi/3 \]

2. **Find the cosine value**:
\[ \cos(4\pi/3) = -\cos(\pi/3) = -\frac{1}{2} \]

3. **Find the principal value**:
\[ \cos^{-1}(-\frac{1}{2}) = \frac{2\pi}{3} \]
Therefore, the principal value is:
\[ \cos^{-1}(\cos(4\pi/3)) = \frac{2\pi}{3} \]

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

1. **Find \( \cos^{-1}(-\frac{1}{2}) \)**:
\[ \cos^{-1}(-\frac{1}{2}) = \frac{2\pi}{3} \]

2. **Calculate the cosine**:
\[ \cos\left(\frac{\pi}{3} + \frac{2\pi}{3}\right) = \cos(\pi) = -1 \]

Thus, the final answer for the third part is:
\[ \cos\left(\frac{\pi}{3} + \cos^{-1}\left(-\frac{1}{2}\right)\right) = -1 \]

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