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

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

• A. \(0\)
• B. \(1\)
• C. \(\frac{1}{2}\)
• D. \(\frac{\sqrt{3}}{2}\)

Answer: (C)

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.

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.