Question

derive 2 pi sin (-pi/4)

Answer 1

To derive \( 2 \pi \sin \left( -\frac{\pi}{4} \right) \), we first need to find the value of \( \sin \left( -\frac{\pi}{4} \right) \).

Using the property of the sine function that states \( \sin(-x) = -\sin(x) \), we can calculate:

\[
\sin \left( -\frac{\pi}{4} \right) = -\sin \left( \frac{\pi}{4} \right)
\]

The value of \( \sin \left( \frac{\pi}{4} \right) \) is \( \frac{\sqrt{2}}{2} \). Therefore:

\[
\sin \left( -\frac{\pi}{4} \right) = -\frac{\sqrt{2}}{2}
\]

Now, substituting this back into the equation:

\[
2 \pi \sin \left( -\frac{\pi}{4} \right) = 2 \pi \left( -\frac{\sqrt{2}}{2} \right)
\]

Simplifying this expression gives:

\[
2 \pi \cdot -\frac{\sqrt{2}}{2} = -\pi \sqrt{2}
\]

Thus, the final result is:

\[
2 \pi \sin \left( -\frac{\pi}{4} \right) = -\pi \sqrt{2}
\]