Certainly! Let's solve the problem step-by-step.
We are given that [tex]\(\cos 30^{\circ} = \frac{\sqrt{3}}{2}\)[/tex].
We need to find [tex]\(\sin 60^{\circ}\)[/tex].
First, we use the complementary angle identity in trigonometry, which states:
[tex]\[
\sin(90^{\circ} - \theta) = \cos(\theta)
\][/tex]
For [tex]\(\theta = 30^{\circ}\)[/tex]:
[tex]\[
\sin(90^{\circ} - 30^{\circ}) = \cos(30^{\circ})
\][/tex]
This simplifies to:
[tex]\[
\sin(60^{\circ}) = \cos(30^{\circ})
\][/tex]
Since we are given that:
[tex]\[
\cos(30^{\circ}) = \frac{\sqrt{3}}{2}
\][/tex]
We can substitute this value into our equation:
[tex]\[
\sin(60^{\circ}) = \frac{\sqrt{3}}{2}
\][/tex]
Therefore, the value of [tex]\(\sin 60^{\circ}\)[/tex] is:
[tex]\[
\sin 60^{\circ} = \frac{\sqrt{3}}{2}
\][/tex]
So, the correct answer to the question is:
[tex]\[
\frac{\sqrt{3}}{2}, \text{ because the angles are complementary}
\][/tex]
This matches the multiple-choice option:
[tex]\(\frac{\sqrt{3}}{2}\)[/tex], because the angles are complementary.
