Question
Note: Enter your answer and show all the steps that you use to solve this problem in the space provided.
Show that sin(x+π)=−sin x
Show all of your work. Answer in 2-3 sentences
Show that sin(x+π)=−sin x
Show all of your work. Answer in 2-3 sentences
Answers
Answered by
GPT-4o mini
To show that \(\sin(x + \pi) = -\sin x\), we can use the sine addition formula:
\[
\sin(a + b) = \sin a \cos b + \cos a \sin b
\]
In this case, let \(a = x\) and \(b = \pi\). Thus,
\[
\sin(x + \pi) = \sin x \cos \pi + \cos x \sin \pi
\]
We know that \(\cos \pi = -1\) and \(\sin \pi = 0\). Substituting these values gives us:
\[
\sin(x + \pi) = \sin x \cdot (-1) + \cos x \cdot 0 = -\sin x
\]
Thus, we have shown that \(\sin(x + \pi) = -\sin x\).
\[
\sin(a + b) = \sin a \cos b + \cos a \sin b
\]
In this case, let \(a = x\) and \(b = \pi\). Thus,
\[
\sin(x + \pi) = \sin x \cos \pi + \cos x \sin \pi
\]
We know that \(\cos \pi = -1\) and \(\sin \pi = 0\). Substituting these values gives us:
\[
\sin(x + \pi) = \sin x \cdot (-1) + \cos x \cdot 0 = -\sin x
\]
Thus, we have shown that \(\sin(x + \pi) = -\sin x\).