`"cos"^(-1)("cos"((7pi)/(5)))=`

Options

A. `(3pi)/(5)`
B. `(2pi)/(5)`
C. `(-7pi)/(5)`
D. `(7pi)/(5)`

Answer

To solve the problem \( \cos^{-1}(\cos(\frac{7\pi}{5})) \), we can follow these steps:

### Step 1: Rewrite the angle
We start by rewriting \( \frac{7\pi}{5} \) in a more manageable form. We can express it as:
\[ \frac{7\pi}{5} = \pi + \frac{2\pi}{5} \]

### Step 2: Use cosine properties
Using the property of cosine, we know that:
\[ \cos(\pi + \theta) = -\cos(\theta) \]
Thus, we can rewrite:
\[ \cos\left(\frac{7\pi}{5}\right) = \cos\left(\pi + \frac{2\pi}{5}\right) = -\cos\left(\frac{2\pi}{5}\right) \]

### Step 3: Substitute into the inverse cosine
Now we substitute this back into our original expression:
\[ \cos^{-1}(\cos(\frac{7\pi}{5})) = \cos^{-1}(-\cos(\frac{2\pi}{5})) \]

### Step 4: Use the property of inverse cosine
We can use the property of inverse cosine that states:
\[ \cos^{-1}(-x) = \pi - \cos^{-1}(x) \]
Applying this property, we have:
\[ \cos^{-1}(-\cos(\frac{2\pi}{5})) = \pi - \cos^{-1}(\cos(\frac{2\pi}{5})) \]

### Step 5: Simplify the expression
Since \( \cos^{-1}(\cos(x)) = x \) for \( x \) in the range of \( [0, \pi] \), we can simplify:
\[ \pi - \cos^{-1}(\cos(\frac{2\pi}{5})) = \pi - \frac{2\pi}{5} \]

### Step 6: Calculate the final value
Now we perform the subtraction:
\[ \pi - \frac{2\pi}{5} = \frac{5\pi}{5} - \frac{2\pi}{5} = \frac{3\pi}{5} \]

### Conclusion
Thus, the final answer is:
\[ \cos^{-1}(\cos(\frac{7\pi}{5})) = \frac{3\pi}{5} \]

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