The value of `cos^-1(cos(-6))+sin^-1(sin5)-tan^-1(tan2)` is

Options

A. `pi-1`
B. `1-pi`
C. `3-pi`
D. `pi-3`

Answer

To solve the expression \( \cos^{-1}(\cos(-6)) + \sin^{-1}(\sin(5)) - \tan^{-1}(\tan(2)) \), we will evaluate each term step by step.

### Step 1: Evaluate \( \cos^{-1}(\cos(-6)) \)

The cosine function is periodic with a period of \( 2\pi \). Thus, we can simplify \( -6 \) by adding \( 2\pi \) until we get a value within the range of \( [0, 2\pi] \).

\[ -6 + 2\pi \approx -6 + 6.2832 \approx 0.2832 \]

Now, since \( \cos^{-1}(x) \) gives us the angle in the range \( [0, \pi] \), we need to find the equivalent angle in this range:

\[ \cos^{-1}(\cos(-6)) = \cos^{-1}(\cos(0.2832)) = 0.2832 \]

### Step 2: Evaluate \( \sin^{-1}(\sin(5)) \)

The sine function is also periodic with a period of \( 2\pi \). To find \( \sin(5) \), we can use the fact that \( 5 \) is already in the range \( [0, 2\pi] \). However, we need to find the equivalent angle in the range \( [-\frac{\pi}{2}, \frac{\pi}{2}] \) for \( \sin^{-1} \).

Since \( 5 \) is greater than \( \frac{\pi}{2} \), we can find its reference angle:

\[ 5 - \pi \approx 5 - 3.1416 \approx 1.8584 \]

Thus,

\[ \sin^{-1}(\sin(5)) = \sin^{-1}(\sin(5 - \pi)) = 5 - \pi \]

### Step 3: Evaluate \( -\tan^{-1}(\tan(2)) \)

The tangent function is periodic with a period of \( \pi \). The angle \( 2 \) is already in the range \( (-\frac{\pi}{2}, \frac{\pi}{2}) \), so:

\[ \tan^{-1}(\tan(2)) = 2 \]

Thus,

\[ -\tan^{-1}(\tan(2)) = -2 \]

### Step 4: Combine all the results

Now we can combine all the evaluated parts:

\[ \cos^{-1}(\cos(-6)) + \sin^{-1}(\sin(5)) - \tan^{-1}(\tan(2) = 0.2832 + (5 - \pi) - 2 \]

Substituting the approximate value of \( \pi \approx 3.1416 \):

\[ = 0.2832 + 5 - 3.1416 - 2 \]
\[ = 0.2832 + 5 - 3.1416 - 2 = 0.2832 + 0.8584 = 1.1416 \]

### Final Answer

Thus, the value of the expression is approximately:

\[ \boxed{1.1416} \]