Question

How do I find the value of $\sin \left(\frac{5\pi }{6}\right)$.

Answer 1

Calculate the value of $\sin \left(\frac{5\pi }{6}\right)$:

First, determine the sign of $\sin \left(\frac{5\pi }{6}\right)$.

$\frac{5\mathrm{\pi }}{6}$ can be rewritten as $\frac{5\mathrm{\pi }}{6}=\frac{\mathrm{\pi }}{2}+\frac{\mathrm{\pi }}{3}$.

Thus $\frac{5\pi }{6}$ belongs to the second quadrant.

It is known that the values of sines are positive $\left(+\right)$ in the second quadrant.

It is also known that, $\mathbf{sin}\mathbf{(}\mathbf{\pi }\mathbf{-}\mathbf{x}\mathbf{)}\mathbf{=}\mathbf{sin}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$.

Thus, $\sin \left(\frac{5\mathrm{\pi }}{6}\right)=\sin \left(\mathrm{\pi }-\frac{\mathrm{\pi }}{6}\right)=\sin \left(\frac{\mathrm{\pi }}{6}\right)=\frac{1}{2}$.

Therefore, the value of $\sin \left(\frac{5\pi }{6}\right)$ is $\frac{1}{2}$.