The principal value of the expression : `cos^(-1) [ cos (-680^(@))] ` is :

Options

A. `(2pi)/9`
B. `(-2pi)/9`
C. `(34pi)/9`
D. `pi/9`

Answer

To find the principal value of the expression \( \cos^{-1} [ \cos (-680^\circ) ] \), we will follow these steps:

### Step 1: Simplify the angle
First, we need to simplify the angle \(-680^\circ\). Since angles can be expressed in terms of their coterminal angles, we can add \(360^\circ\) until the angle is within the standard range of \(0^\circ\) to \(360^\circ\).

\[ -680^\circ + 2 \times 360^\circ = -680^\circ + 720^\circ = 40^\circ \]

### Step 2: Evaluate the cosine
Now we can evaluate the cosine of the simplified angle:

\[ \cos(-680^\circ) = \cos(40^\circ) \]

### Step 3: Apply the inverse cosine function
Next, we apply the inverse cosine function to the result from Step 2:

\[ \cos^{-1} [ \cos(40^\circ) ] \]

### Step 4: Determine the principal value
The principal value of \( \cos^{-1} [ \cos(x) ] \) is \( x \) when \( x \) is within the range \( [0, \pi] \) (or \( [0^\circ, 180^\circ] \)). Since \( 40^\circ \) is within this range, we have:

\[ \cos^{-1} [ \cos(40^\circ) ] = 40^\circ \]

### Final Answer
Thus, the principal value of the expression \( \cos^{-1} [ \cos (-680^\circ) ] \) is:

\[ \boxed{40^\circ} \]

---