The expression \(\sin(180^\circ + c)\) can be simplified using the angle addition identity for sine:

\[
\sin(180^\circ + c) = \sin 180^\circ \cos c + \cos 180^\circ \sin c
\]

We know that:

\[
\sin 180^\circ = 0 \quad \text{and} \quad \cos 180^\circ = -1
\]

Substituting these values:

\[
\sin(180^\circ + c) = 0 \cdot \cos c + (-1) \cdot \sin c
\]

Simplifying this, we get:

\[
\sin(180^\circ + c) = -\sin c
\]

So, the simplified form of \(\sin(180^\circ + c)\) is \(-\sin c\).

Would you like further details or have any questions?

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Here are 5 related questions to deepen your understanding:
1. How would you simplify \(\cos(180^\circ + c)\)?
2. What is the value of \(\sin(360^\circ + c)\)?
3. How does the sine function behave under different angle transformations like \(\sin(90^\circ + c)\)?
4. What is the significance of the angle \(180^\circ\) in trigonometry?
5. Can you derive the general formula for \(\sin(180^\circ - c)\)?

**Tip:** Memorizing key trigonometric identities, especially those involving standard angles like \(180^\circ\), \(90^\circ\), and \(360^\circ\), can greatly simplify the process of solving trigonometric expressions.

Learn how to simplify the expression \( \sin(180^\circ + c) \) using trigonometric identities. This problem involves applying the angle addition identity for sine and understanding the behavior of sine and cosine functions at specific angles. Suitable for high school students studying trigonometry.

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