The side lengths of a [tex][tex]$30-60-90$[/tex][/tex] triangle are in the ratio [tex][tex]$1: \sqrt{3}: 2$[/tex][/tex]. What is [tex][tex]$\sin 30^{\circ}$[/tex][/tex]?

A. [tex][tex]$\sqrt{3}$[/tex][/tex]
B. [tex][tex]$\frac{1}{2}$[/tex][/tex]
C. [tex][tex]$\frac{\sqrt{3}}{2}$[/tex][/tex]
D. [tex][tex]$\frac{\sqrt{3}}{3}$[/tex][/tex]



Answer :

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].