sin(x/2)-sin($$ \frac{\pi}{2}-\frac{3x}{2})=0 $$

To solve the equation:

\[
\sin\left(\frac{x}{2}\right) - \sin\left(\frac{\pi}{2} - \frac{3x}{2}\right) = 0
\]

we can use the trigonometric identity:

\[
\sin(A) - \sin(B) = 2 \cos\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right)
\]

Let's set $ A = \frac{x}{2} $ and $ B = \frac{\pi}{2} - \frac{3x}{2} $.

Applying the identity:

\[
2 \cos\left(\frac{\frac{x}{2} + \left(\frac{\pi}{2} - \frac{3x}{2}\right)}{2}\right) \sin\left(\frac{\frac{x}{2} - \left(\frac{\pi}{2} - \frac{3x}{2}\right)}{2}\right) = 0
\]

Simplify the arguments of the cosine and sine functions:

For the cosine term:

\[
\frac{\frac{x}{2} + \left(\frac{\pi}{2} - \frac{3x}{2}\right)}{2} = \frac{\frac{x}{2} + \frac{\pi}{2} - \frac{3x}{2}}{2} = \frac{\frac{\pi}{2} - x}{2} = \frac{\pi}{4} - \frac{x}{2}
\]

So, the cosine term becomes:

\[
\cos\left(\frac{\pi}{4} - \frac{x}{2}\right)
\]

For the sine term:

\[
\frac{\frac{x}{2} - \left(\frac{\pi}{2} - \frac{3x}{2}\right)}{2} = \frac{\frac{x}{2} - \frac{\pi}{2} + \frac{3x}{2}}{2} = \frac{2x - \frac{\pi}{2}}{2} = x - \frac{\pi}{4}
\]

So, the sine term becomes:

\[
\sin\left(x - \frac{\pi}{4}\right)
\]

Now, the equation becomes:

\[
2 \cos\left(\frac{\pi}{4} - \frac{x}{2}\right) \sin\left(x - \frac{\pi}{4}\right) = 0
\]

This product equals zero if either factor equals zero:

1. $ \cos\left(\frac{\pi}{4} - \frac{x}{2}\right) = 0 $
2. $ \sin\left(x - \frac{\pi}{4}\right) = 0 $

### Solving for $ \cos\left(\frac{\pi}{4} - \frac{x}{2}\right) = 0 $:

Cosine equals zero at:

\[
\frac{\pi}{4} - \frac{x}{2} = \frac{\pi}{2} + n\pi \quad \text{for integer } n
\]

Solve for $ x $:

\[
\frac{\pi}{4} - \frac{x}{2} = \frac{\pi}{2} + n\pi
\]
\[
-\frac{x}{2} = \frac{\pi}{2} + n\pi - \frac{\pi}{4}
\]
\[
-\frac{x}{2} = \frac{\pi}{4} + n\pi
\]
\[
x = -\frac{\pi}{2} - 2n\pi
\]

### Solving for $ \sin\left(x - \frac{\pi}{4}\right) = 0 $:

Sine equals zero at:

\[
x - \frac{\pi}{4} = m\pi \quad \text{for integer } m
\]

Solve for $ x $:

\[
x = \frac{\pi}{4} + m\pi
\]

### Final Solution:

The general solutions are:

1. $ x = -\frac{\pi}{2} - 2n\pi $
2. $ x = \frac{\pi}{4} + m\pi $

Would you like further details or have any questions?

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Here are 5 related questions to expand on this topic:

1. How do trigonometric identities simplify solving trigonometric equations?
2. What are the general solutions for $ \cos(\theta) = 0 $ and $ \sin(\theta) = 0 $?
3. Can you solve the equation $ \cos(x) + \sin(x) = 0 $?
4. How would the solutions change if the equation were $ \sin\left(\frac{x}{2}\right) + \sin\left(\frac{\pi}{2} - \frac{3x}{2}\right) = 0 $?
5. What are the applications of solving trigonometric equations in real-world problems?

**Tip:** When dealing with trigonometric equations, knowing key identities like the sum-to-product identities can greatly simplify the problem-solving process.

Explore how to solve the trigonometric equation sin(x/2) - sin(pi/2 - 3x/2) = 0 step-by-step. This problem involves using trigonometric identities, specifically the sum-to-product identities, to find general solutions. Suitable for advanced high school students and above.

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