The value of `cos^(-1)(cos 5pi/3)+sin^(-1) ("sin" (5pi)/3)` is :

Options

A. `pi/2`
B. `(5pi)/3`
C. `(10pi)/3`
D. 0

Answer

To solve the expression \( \cos^{-1}(\cos(5\pi/3)) + \sin^{-1}(\sin(5\pi/3)) \), we will break it down step by step.

### Step 1: Simplify \( \sin^{-1}(\sin(5\pi/3)) \)

First, we need to find the value of \( 5\pi/3 \) in terms of a standard angle.

- \( 5\pi/3 \) is equivalent to \( 2\pi - \pi/3 \), which means it lies in the fourth quadrant.

In the fourth quadrant, the sine function is negative. Therefore, we can express:

\[ \sin(5\pi/3) = \sin(2\pi - \pi/3) = -\sin(\pi/3) \]

Since \( \sin(\pi/3) = \frac{\sqrt{3}}{2} \), we have:

\[ \sin(5\pi/3) = -\frac{\sqrt{3}}{2} \]

Now, substituting this into the inverse sine function:

\[ \sin^{-1}(\sin(5\pi/3)) = \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right) \]

The principal value of \( \sin^{-1}(x) \) is in the range \([- \frac{\pi}{2}, \frac{\pi}{2}]\). Therefore,

\[ \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right) = -\frac{\pi}{3} \]

### Step 2: Simplify \( \cos^{-1}(\cos(5\pi/3)) \)

Next, we simplify \( \cos^{-1}(\cos(5\pi/3)) \).

- Again, since \( 5\pi/3 = 2\pi - \pi/3 \), we can write:

\[ \cos(5\pi/3) = \cos(2\pi - \pi/3) = \cos(\pi/3) \]

Since \( \cos(\pi/3) = \frac{1}{2} \), we have:

\[ \cos(5\pi/3) = \frac{1}{2} \]

Now, substituting this into the inverse cosine function:

\[ \cos^{-1}(\cos(5\pi/3)) = \cos^{-1}\left(\frac{1}{2}\right) \]

The principal value of \( \cos^{-1}(x) \) is in the range \([0, \pi]\). Therefore,

\[ \cos^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{3} \]

### Step 3: Combine the Results

Now we can combine the results from Steps 1 and 2:

\[ \cos^{-1}(\cos(5\pi/3)) + \sin^{-1}(\sin(5\pi/3)) = \frac{\pi}{3} + \left(-\frac{\pi}{3}\right) \]

This simplifies to:

\[ \frac{\pi}{3} - \frac{\pi}{3} = 0 \]

### Final Answer

Thus, the value of \( \cos^{-1}(\cos(5\pi/3)) + \sin^{-1}(\sin(5\pi/3)) \) is:

\[ \boxed{0} \]