Question

\sin \left(\frac{7\pi }{6}\right)

Answer 1

To find \(\sin\left(\frac{7\pi}{6}\right)\), we first recognize that \(\frac{7\pi}{6}\) is in the third quadrant.

1. **Reference angle**: The reference angle for \(\frac{7\pi}{6}\) can be found by subtracting \(\pi\):
\[
\text{Reference angle} = \frac{7\pi}{6} - \pi = \frac{7\pi}{6} - \frac{6\pi}{6} = \frac{\pi}{6}
\]

2. **Sine function in the third quadrant**: In the third quadrant, the sine function is negative. So we have:
\[
\sin\left(\frac{7\pi}{6}\right) = -\sin\left(\frac{\pi}{6}\right)
\]

3. **Value of \(\sin\left(\frac{\pi}{6}\right)\)**: The sine of \(\frac{\pi}{6}\) is known to be:
\[
\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}
\]

4. **Putting it all together**:
\[
\sin\left(\frac{7\pi}{6}\right) = -\frac{1}{2}
\]

Thus, the final result is:
\[
\sin\left(\frac{7\pi}{6}\right) = -\frac{1}{2}
\]