Prove for any real number x and y sin x = sin y implies x=nπ+(-1)^πy where nEZ

To prove that \( \sin(x) = \sin(y) \) implies \( x = n\pi + (-1)^n y \) for some integer \( n \in \mathbb{Z} \), let us analyze the periodicity and symmetry properties of the sine function.

### Step 1: General solution to \( \sin(x) = \sin(y) \)

The sine function is periodic with period \( 2\pi \), meaning that:

\[
\sin(x) = \sin(x + 2k\pi) \quad \text{for any integer} \, k.
\]

Additionally, the sine function is symmetric, specifically:

\[
\sin(x) = \sin(\pi - x).
\]

Thus, if \( \sin(x) = \sin(y) \), there are two possible cases:

1. \( x = y + 2n\pi \), for some \( n \in \mathbb{Z} \) (due to the periodicity of sine).
2. \( x = \pi - y + 2n\pi \), for some \( n \in \mathbb{Z} \) (due to the symmetry \( \sin(x) = \sin(\pi - x) \)).

### Step 2: Combine these cases

From these two cases, we can write the general solution as:

\[
x = y + 2n\pi \quad \text{or} \quad x = \pi - y + 2n\pi.
\]

The second case can be rewritten as:

\[
x = (2n + 1)\pi - y.
\]

### Step 3: Simplify and express in terms of \( (-1)^n \)

We now combine these two cases into a single equation. Notice that:

- \( x = y + 2n\pi \) corresponds to \( x = n\pi + (-1)^n y \) when \( n \) is even.
- \( x = (2n + 1)\pi - y \) corresponds to \( x = n\pi + (-1)^n y \) when \( n \) is odd.

Thus, the general solution to \( \sin(x) = \sin(y) \) can be written as:

\[
x = n\pi + (-1)^n y, \quad n \in \mathbb{Z}.
\]

### Conclusion:

The equation \( \sin(x) = \sin(y) \) implies that \( x = n\pi + (-1)^n y \) for some integer \( n \in \mathbb{Z} \). This solution accounts for both the periodicity and symmetry properties of the sine function.

Would you like more details on any of these steps? Here are five related questions for deeper exploration:

1. How does the periodicity of the cosine function compare to sine?
2. Can this approach be used to solve \( \cos(x) = \cos(y) \)?
3. What is the geometric interpretation of the symmetry \( \sin(\pi - x) = \sin(x) \)?
4. How would the general solution change for \( \tan(x) = \tan(y) \)?
5. Can you apply this reasoning to find all solutions to \( \sin(x) = 0 \)?

**Tip:** Always consider the symmetry and periodicity of trigonometric functions when solving equations like \( \sin(x) = \sin(y) \).

Explore the proof that sin(x) = sin(y) implies x = nπ + (-1)^n y for real numbers x and y. This solution discusses the periodicity and symmetry properties of the sine function, making it ideal for students in Grades 11-12 studying trigonometry.

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