Which expression is equivalent to sin(7π/6)?

Understand the Problem

The question is asking for an expression that is mathematically equivalent to the sine of the angle 7π/6 radians. This involves understanding the properties of the sine function and possibly using angle identities.

Answer

$-\frac{1}{2}$

Steps to Solve

Identify the angle in standard position
The angle $7\pi/6$ radians is more than $\pi$ radians (or $180^\circ$), so it lies in the third quadrant.
Find a reference angle
To find the reference angle, subtract $\pi$ (or $180^\circ$) from $7\pi/6$:
$$ \text{Reference angle} = 7\pi/6 - \pi = 7\pi/6 - 6\pi/6 = \pi/6 $$
So, the reference angle is $\pi/6$ radians.
Determine the sine value
The sine value in the third quadrant is negative. Therefore, the sine of $7\pi/6$ will be the negative sine of the reference angle:
$$ \sin\left(\frac{7\pi}{6}\right) = -\sin\left(\frac{\pi}{6}\right) $$
Calculate the sine of the reference angle
We know that:
$$ \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} $$
Final expression
Substituting this value back into our equation gives:
$$ \sin\left(\frac{7\pi}{6}\right) = -\frac{1}{2} $$

The expression that is mathematically equivalent to the sine of the angle $7\pi/6$ radians is $-\frac{1}{2}$.

More Information

The sine function is periodic and its values change depending on the quadrant. The key here is recognizing that angles in the third quadrant have negative sine values, and knowing the sine values of special angles helps find their equivalents.

Tips

Misidentifying the quadrant of the angle: It's important to remember that $7\pi/6$ is in the third quadrant where sine is negative.
Forgetting to consider the negative sign when using reference angles in the third quadrant.

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