Question

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Show that sin(x+π)=−sin x
Show all of your work.

Answer 1

To show that \(\sin(x + \pi) = -\sin x\), we will use the sine addition formula and properties of the sine function.

### Step 1: Apply the Sine Addition Formula

The sine addition formula states that:

\[
\sin(a + b) = \sin a \cos b + \cos a \sin b
\]

Let \(a = x\) and \(b = \pi\). Then we can rewrite the expression as:

\[
\sin(x + \pi) = \sin x \cos \pi + \cos x \sin \pi
\]

### Step 2: Substitute Values of \(\cos \pi\) and \(\sin \pi\)

We know that:

\[
\cos \pi = -1 \quad \text{and} \quad \sin \pi = 0
\]

Substituting these values into the equation gives:

\[
\sin(x + \pi) = \sin x \cdot (-1) + \cos x \cdot 0
\]

### Step 3: Simplify the Expression

This simplifies to:

\[
\sin(x + \pi) = -\sin x + 0
\]

Thus, we have:

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

### Conclusion

We have shown that:

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

This completes the proof.