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

Answer & Explanation

mauricio0815sh

Beginner2021-12-12Added 34 answers

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

sirpsta3u

Beginner2021-12-13Added 42 answers

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