To determine [tex]\(\sin 30^{\circ}\)[/tex] for a [tex]\(30-60-90\)[/tex] triangle, we can utilize the known properties of this special type of triangle.
1. Understanding the side lengths: In a [tex]\(30-60-90\)[/tex] triangle, the sides have a specific ratio: [tex]\(1 : \sqrt{3} : 2\)[/tex]. This means:
- The shortest side (opposite the 30-degree angle) is [tex]\(1\)[/tex],
- The longer leg (opposite the 60-degree angle) is [tex]\(\sqrt{3}\)[/tex],
- The hypotenuse (opposite the 90-degree angle) is [tex]\(2\)[/tex].
2. Definition of sine: The sine of an angle in a right triangle is the ratio of the length of the side opposite the angle to the hypotenuse. In this case, for [tex]\(\sin 30^{\circ}\)[/tex]:
- The side opposite the 30-degree angle is [tex]\(1\)[/tex],
- The hypotenuse is [tex]\(2\)[/tex].
3. Calculation of [tex]\(\sin 30^{\circ}\)[/tex]:
[tex]\[
\sin 30^{\circ} = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{2}
\][/tex]
Thus, [tex]\(\sin 30^{\circ}\)[/tex] is [tex]\(\frac{1}{2}\)[/tex].
The answer is [tex]\(B. \frac{1}{2}\)[/tex].
