Question

How do you verify the identity `sin(A+pi)=-sinA`?

Answer 1

There are multiple ways of proving the identity. Please, see below.

Explanation:

First way, probably the more direct method, is by the unit circle:

Poorly created image by me in Paint.

We see that, if we add `pi` to some angle `theta`, or in our case, `A`, `|sin theta| = |sin (theta+pi)|`, but since thery're on opposite parts of the center `O`, which is not in the image, they're of opposite sign.

`color(red)( :. sin(A+pi) = - sinA`

Another method to verify our relation is by applying the sum formula for sine:

`sin(a+b) = sinacosb+cosasinb`

`=> sin(A+pi) = sinAcospi + cosAsinpi=-sinA`

`color(red)( :. sin(A+pi)=-sinA`