To determine [tex]\(\sin 60^\circ\)[/tex], we can refer to a well-known value from trigonometry.
Let's work through the steps to find [tex]\(\sin 60^\circ\)[/tex]:
1. Understanding the Unit Circle and Special Angles:
The angle 60 degrees (or [tex]\(\pi/3\)[/tex] radians) is one of the special angles in trigonometry. Knowing the sine values for these angles can be helpful.
2. Special Triangles:
In a 30-60-90 triangle, the sides have a specific ratio. If we take a right triangle with angles 30 degrees, 60 degrees, and 90 degrees:
- The side opposite the 30-degree angle is the smallest and we'll call its length [tex]\(a\)[/tex].
- The side opposite the 60-degree angle is [tex]\(\sqrt{3}a\)[/tex].
- The hypotenuse (opposite the 90-degree angle) is [tex]\(2a\)[/tex].
3. Calculating [tex]\(\sin 60^\circ\)[/tex]:
In our 30-60-90 triangle:
- The sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.
- For 60 degrees:
[tex]\[
\sin 60^\circ = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\sqrt{3}a}{2a} = \frac{\sqrt{3}}{2}
\][/tex]
4. Verification:
Checking the numerical value of [tex]\(\frac{\sqrt{3}}{2}\)[/tex]:
- [tex]\(\sqrt{3} \approx 1.732\)[/tex]
- So, [tex]\(\frac{1.732}{2} \approx 0.866\)[/tex]
Therefore, the value of [tex]\(\sin 60^\circ\)[/tex] is approximately [tex]\(0.866\)[/tex], and the corresponding option is B. Hence,
[tex]\[
\sin 60^\circ = \frac{\sqrt{3}}{2}
\][/tex]
