Question

Note: Enter your answer and show all the steps that you use to solve this problem in the spaceprovided.Show that sin (x +pi ) = - sin xShow all of your work

Answer 1

ANSWER and EXPLANATION

We want to prove that:

`\sin(x+\pi)=-\sin x`

Let us start with the left-hand side of the equation.

Using trigonometric identities for sine, we have that:

`\sin(A+B)=\sin A\cos B+\sin B\cos A`

Applying this identity to the left-hand side of the equation:

`\sin(x+\pi)=\sin x\cos\pi+\sin\pi\cos x`

We know that:

`\begin{gathered} \cos\pi=-1 \\ \sin\pi=0 \end{gathered}`

Substituting those values into the above expression:

`\begin{gathered} \sin x(-1)+(0)\cos x \\ \Rightarrow-\sin x \end{gathered}`

Since the left-hand side of the equation is equal to the right-hand side, we have that it has been proven.