Verify the identity sin⁡(A+π)=−sin⁡A

Question

mauricio0815sh · Accepted Answer

Use the Sum formula for sin:    sin⁡(a+b)=sin⁡αcos⁡β+cos⁡αsin⁡β    ⇒sin⁡(A+π)=sin⁡Acos⁡π+cos⁡Asin⁡π=−sin⁡A    ∴sin⁡(A+π)=−sin⁡A

sirpsta3u · Accepted Answer

You can prove it using unit circle:
If we add

π

to some angle

θ

,

\sin θ = \sin (θ + π)

, but since they are on opposite parts of the center O, they have an opposite sign:

\sin (A + π) = - \sin A

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