What is \( \sin(30^\circ) \)?

The sine function gives the ratio of the opposite side to the hypotenuse in a right triangle or can be interpreted as the y-coordinate of a point on the unit circle. For an angle of (30^\circ), which corresponds to (\frac{\pi}{6}) radians, it is a well-known fact in trigonometry that the sine value is (\frac{1}{2}).

This can be visualized using a 30-60-90 triangle. In such a triangle, the sides are in the ratio (1 : \sqrt{3} : 2). The side opposite the (30^\circ) angle is (1) (the shortest leg), and the hypotenuse is (2). Therefore, the sine of (30^\circ) is:

[

\sin(30^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{2}

]

This established result makes (\frac{1}{2}) the correct value for (\sin(30^\circ)). Understanding this relationship is essential in trigonometry and helps in solving various problems across different levels of mathematics.