Answer :
To find the value of [tex]\(\sin 30^{\circ}\)[/tex], we can use our knowledge of trigonometry and the unit circle.
1. Understanding the unit circle: The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. The angle [tex]\(\theta\)[/tex] forms by a radius line and the positive x-axis, measured counterclockwise.
2. Reference angles and special triangles: In a unit circle, specific angles have well-known sine and cosine values. These angles often correspond to special right triangles. For example, the 30-60-90 triangle, which is derived from an equilateral triangle, is significant in trigonometry.
- In a 30-60-90 triangle, the sides are in the ratio [tex]\(1 : \sqrt{3} : 2\)[/tex]. The 30° angle is opposite the side with length 1, the 60° angle is opposite the side with length [tex]\(\sqrt{3}\)[/tex], and the hypotenuse is 2.
3. Calculating [tex]\(\sin 30^{\circ}\)[/tex]: The sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse. For a 30° angle in a 30-60-90 triangle, the side opposite 30° has length 1, and the hypotenuse has length 2.
[tex]\[ \sin 30^{\circ} = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{2} \][/tex]
Thus, [tex]\(\sin 30^{\circ} = \frac{1}{2}\)[/tex].
The correct answer is:
[tex]\[ \text{B. } \frac{1}{2} \][/tex]
1. Understanding the unit circle: The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. The angle [tex]\(\theta\)[/tex] forms by a radius line and the positive x-axis, measured counterclockwise.
2. Reference angles and special triangles: In a unit circle, specific angles have well-known sine and cosine values. These angles often correspond to special right triangles. For example, the 30-60-90 triangle, which is derived from an equilateral triangle, is significant in trigonometry.
- In a 30-60-90 triangle, the sides are in the ratio [tex]\(1 : \sqrt{3} : 2\)[/tex]. The 30° angle is opposite the side with length 1, the 60° angle is opposite the side with length [tex]\(\sqrt{3}\)[/tex], and the hypotenuse is 2.
3. Calculating [tex]\(\sin 30^{\circ}\)[/tex]: The sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse. For a 30° angle in a 30-60-90 triangle, the side opposite 30° has length 1, and the hypotenuse has length 2.
[tex]\[ \sin 30^{\circ} = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{2} \][/tex]
Thus, [tex]\(\sin 30^{\circ} = \frac{1}{2}\)[/tex].
The correct answer is:
[tex]\[ \text{B. } \frac{1}{2} \][/tex]
