`cos^(-1)[cos(- 17/15 pi )]` is equal to

Options

A. `-(17pi)/15`
B. `(17pi)/15`
C. `(2pi)/15`
D. `(13pi)/15`

Answer

To solve the expression \( \cos^{-1}[\cos(-\frac{17}{15} \pi)] \), we can follow these steps:

### Step 1: Simplify the angle
The first step is to simplify the angle \(-\frac{17}{15} \pi\). We can express this angle in a more manageable form by adding \(2\pi\) (which is a full rotation) until the angle is within the range of \(0\) to \(2\pi\).

\[ -\frac{17}{15} \pi + 2\pi = -\frac{17}{15} \pi + \frac{30}{15} \pi = \frac{13}{15} \pi \]

### Step 2: Use the cosine function
Now, we can rewrite the original expression using this simplified angle:

\[ \cos^{-1}[\cos(-\frac{17}{15} \pi)] = \cos^{-1}[\cos(\frac{13}{15} \pi)] \]

### Step 3: Apply the property of cosine
The property of the cosine function states that \( \cos(\pi + \theta) = -\cos(\theta) \). Here, \(\frac{13}{15} \pi\) is in the second quadrant, where cosine is negative.

To find the equivalent angle in the range of \(0\) to \(\pi\), we can use the fact that:

\[ \cos(\pi - \theta) = -\cos(\theta) \]

Thus, we can express \(\frac{13}{15} \pi\) as:

\[ \frac{13}{15} \pi = \pi - \frac{2}{15} \pi \]

### Step 4: Evaluate the inverse cosine
Now we can evaluate:

\[ \cos^{-1}[\cos(\frac{13}{15} \pi)] = \pi - \frac{2}{15} \pi \]

### Step 5: Final calculation
Calculating this gives:

\[ \pi - \frac{2}{15} \pi = \frac{15}{15} \pi - \frac{2}{15} \pi = \frac{13}{15} \pi \]

Thus, the final answer is:

\[ \cos^{-1}[\cos(-\frac{17}{15} \pi)] = \frac{13}{15} \pi \]

### Summary of Steps
1. Simplify the angle by adding \(2\pi\).
2. Rewrite the expression using the simplified angle.
3. Use properties of cosine to find the equivalent angle.
4. Evaluate the inverse cosine.
5. Calculate the final result.