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The question asks for the value of \( k \) in the equation:

\[
\arcsin\left(\frac{1}{\sqrt{2}}\right) = k \cdot \pi
\]

We know from trigonometric values that:

\[
\arcsin\left(\frac{1}{\sqrt{2}}\right) = \frac{\pi}{4}
\]

This corresponds to a 45° angle in radians. Thus, we can equate this to the expression on the right side of the equation:

\[
k \cdot \pi = \frac{\pi}{4}
\]

To find \( k \), divide both sides by \( \pi \):

\[
k = \frac{1}{4}
\]

So, the value of \( k \) is:

\[
k = \frac{1}{4}
\]

Let me know if you would like more details, or if you have any questions!

Here are five related questions:
1. What is the general solution for \( \arcsin(x) \)?
2. Can \( \arcsin(x) \) be expressed in terms of other trigonometric functions?
3. What is the domain and range of the \( \arcsin \) function?
4. How would the equation change if \( \arcsin(-x) \) were used?
5. How does the unit circle help in understanding inverse trigonometric functions?

**Tip:** The inverse sine function \( \arcsin(x) \) returns an angle, so it's important to know key angle values in radians.

Følgende værdi af arcusinus, kan skrives som et rationelt tal k, multipliceret med π: arcsin(1/√2) = k · π. Hvad er værdien af k?

Solve the trigonometric equation arcsin(1/√2) = k · π and find the value of k. This problem involves inverse trigonometric functions and radians, suitable for high school students in Grades 9-12.

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