Is sin(pi)=sin(-pi) ?

The answer to whether sin(pi)=sin(-pi) is True

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prove sin(pi)=sin(-pi)

prove

sin (π) = sin (- π)

True

Solution steps

sin (π) = sin (- π)

Manipulating left side

sin (π)

Simplify

= 0

Manipulating right side

sin (- π)

Use the negative angle identity:

sin (- x) = - sin (x)

= - sin (π)

Simplify

- sin (π) : 0

- sin (π)

Use the following trivial identity:

sin (π) = 0

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= - 0

= 0

= 0

We showed that the two sides could take the same form

\Rightarrow True