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 follow these steps:

### Step 1: Simplify \( 5\pi/3 \)
First, we need to simplify \( 5\pi/3 \) to find its equivalent angle within the range of \( [0, 2\pi) \):
\[ 5\pi/3 = 2\pi - \pi/3 \]
This means \( 5\pi/3 \) is equivalent to \( -\pi/3 \) when considering the unit circle.

### Step 2: Calculate \( \cos(5\pi/3) \)
Using the cosine function:
\[ \cos(5\pi/3) = \cos(2\pi - \pi/3) = \cos(\pi/3) = \frac{1}{2} \]

### Step 3: Calculate \( \sin(5\pi/3) \)
Using the sine function:
\[ \sin(5\pi/3) = \sin(2\pi - \pi/3) = -\sin(\pi/3) = -\frac{\sqrt{3}}{2} \]

### Step 4: Evaluate \( \cos^{-1}(\cos(5\pi/3)) \)
Now we can evaluate \( \cos^{-1}(\cos(5\pi/3)) \):
\[ \cos^{-1}(\cos(5\pi/3)) = \cos^{-1}\left(\frac{1}{2}\right) \]
Since \( \cos^{-1}(x) \) gives us the angle in the range \( [0, \pi] \), we find:
\[ \cos^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{3} \]

### Step 5: Evaluate \( \sin^{-1}(\sin(5\pi/3)) \)
Next, we evaluate \( \sin^{-1}(\sin(5\pi/3)) \):
\[ \sin^{-1}(\sin(5\pi/3)) = \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right) \]
Since \( \sin^{-1}(x) \) gives us the angle in the range \( [-\frac{\pi}{2}, \frac{\pi}{2}] \), we find:
\[ \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right) = -\frac{\pi}{3} \]

### Step 6: Combine the results
Now we can combine the results from Steps 4 and 5:
\[ \cos^{-1}(\cos(5\pi/3)) + \sin^{-1}(\sin(5\pi/3)) = \frac{\pi}{3} - \frac{\pi}{3} = 0 \]

### Final Answer
Thus, the value of \( \cos^{-1}(\cos(5\pi/3)) + \sin^{-1}(\sin(5\pi/3)) \) is:
\[ \boxed{0} \]