amolent3u

2021-12-11

Verify the identity $\mathrm{sin}\left(A+\pi \right)=-\mathrm{sin}A$

mauricio0815sh

Use the Sum formula for sin:
$\mathrm{sin}\left(a+b\right)=\mathrm{sin}\alpha \mathrm{cos}\beta +\mathrm{cos}\alpha \mathrm{sin}\beta$
$⇒\mathrm{sin}\left(A+\pi \right)=\mathrm{sin}A\mathrm{cos}\pi +\mathrm{cos}A\mathrm{sin}\pi =-\mathrm{sin}A$
$\therefore \mathrm{sin}\left(A+\pi \right)=-\mathrm{sin}A$

sirpsta3u

You can prove it using unit circle:
If we add $\pi$ to some angle $\theta$, $\mathrm{sin}\theta =\mathrm{sin}\left(\theta +\pi \right)$, but since they are on opposite parts of the center O, they have an opposite sign:
$\mathrm{sin}\left(A+\pi \right)=-\mathrm{sin}A$

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