To solve \( \arcsin(\cos(7\pi/5)) \), follow these steps:

### Step 1: Simplify \( \cos(7\pi/5) \)

We can express \( 7\pi/5 \) in terms of a simpler angle by noting that:

\[
7\pi/5 = \pi + 2\pi/5
\]

This places the angle in the third quadrant, where cosine is negative. Since the cosine function is even, we can use:

\[
\cos(7\pi/5) = -\cos(2\pi/5)
\]

We now have \( \cos(7\pi/5) = -\cos(2\pi/5) \), but we don't need the exact value of \( \cos(2\pi/5) \) for the purpose of solving the arcsin function.

### Step 2: Analyze \( \arcsin \) range

The range of the arcsine function, \( \arcsin(x) \), is \( [-\pi/2, \pi/2] \). So the result must lie within this interval.

### Step 3: Find \( \arcsin(-\cos(2\pi/5)) \)

Since \( \cos(2\pi/5) \) is a positive value, the arcsine of \( -\cos(2\pi/5) \) will give a negative angle. Let's call the value \( \theta \), where:

\[
\theta = -\arcsin(\cos(2\pi/5))
\]

Since \( \arcsin(\cos(2\pi/5)) \) is a positive angle less than \( \pi/2 \), the solution to the problem is a negative value of that angle.

Thus:

\[
\arcsin(\cos(7\pi/5)) = -\arcsin(\cos(2\pi/5))
\]

This gives the exact value in terms of the arcsine of \( \cos(2\pi/5) \).

Would you like to dive deeper into calculating these values?

### Related Questions:
1. What are the exact values of \( \cos(2\pi/5) \)?
2. How is the range of the arcsin function determined?
3. What is the difference between arcsin and arccos?
4. How can you determine the quadrant of any trigonometric function?
5. What are some real-world applications of inverse trigonometric functions?

**Tip:** Always check the quadrant when dealing with trigonometric functions to avoid sign errors.

Discover the exact value of arcsin(cos(7π/5)) with a step-by-step solution. This problem explores key trigonometric concepts, including cosine transformations, inverse sine properties, and the unit circle. Suitable for students in Grades 10-12.

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