i dont understand why the answer is also x=m pi divided by 2

To explain why the answer involves \( x = \frac{m\pi}{2} \), let's break down the context in which such a solution typically appears. This kind of expression often arises in trigonometric equations, particularly those involving sine or cosine functions. For example, consider an equation like:

\[
\sin(x) = 0
\]

The sine function equals zero at specific points, namely:

\[
x = n\pi \quad \text{for integer values of } n.
\]

However, if the equation involves a cosine function, such as:

\[
\cos(x) = 0,
\]

the solution set changes to:

\[
x = \frac{(2m+1)\pi}{2} \quad \text{for integer values of } m.
\]

This is because the cosine function equals zero at \( \frac{\pi}{2} \), \( \frac{3\pi}{2} \), etc., which can be written as \( x = \frac{(2m+1)\pi}{2} \).

If you encountered \( x = \frac{m\pi}{2} \) specifically, it suggests you might be dealing with a different equation or a specific condition within a trigonometric context where:

\[
\sin(x) = \pm 1
\]

This occurs at \( x = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \) etc., which can be generalized as \( x = \frac{(2m+1)\pi}{2} \).

To summarize, \( x = \frac{m\pi}{2} \) is a general form of solutions where \( x \) is spaced by \( \frac{\pi}{2} \), and \( m \) is an integer representing different multiples of that spacing.

Would you like further details, or do you have any specific questions about this explanation?

### Related Questions:
1. What is the general solution to the equation \( \sin(x) = 0 \)?
2. How do you derive the general solution for \( \cos(x) = 0 \)?
3. Can you explain why trigonometric functions have periodic solutions?
4. What are the specific values of \( m \) that satisfy \( \cos(x) = 0 \)?
5. How does the general solution for \( \sin(x) = 1 \) differ from \( \sin(x) = 0 \)?

### Tip:
When solving trigonometric equations, always consider the periodic nature of the sine and cosine functions, which often lead to multiple valid solutions.

Explore why solutions involving x = mπ/2 occur in trigonometric equations. This explanation covers the periodic nature of trigonometric functions and how solutions are derived, focusing on cosine functions. Suitable for advanced high school students and beyond.

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