To solve for [tex]\(\sin 60^\circ\)[/tex] given that [tex]\(\cos 30^\circ = \frac{\sqrt{3}}{2}\)[/tex], we can use the fact that the angles are complementary. Complementary angles are two angles that add up to 90 degrees. In trigonometry, the sine of one angle in a right triangle is equal to the cosine of its complementary angle. Hence,
[tex]\[ \sin (90^\circ - \theta) = \cos \theta \][/tex]
For [tex]\(\theta = 30^\circ\)[/tex]:
[tex]\[ \sin 60^\circ = \sin (90^\circ - 30^\circ) = \cos 30^\circ \][/tex]
We are given that:
[tex]\[ \cos 30^\circ = \frac{\sqrt{3}}{2} \][/tex]
Therefore:
[tex]\[ \sin 60^\circ = \frac{\sqrt{3}}{2} \][/tex]
Converting [tex]\(\frac{\sqrt{3}}{2}\)[/tex] to its decimal form, we get approximately 0.8660254037844386. So, the correct answer is:
[tex]\(\sin 60^\circ = \frac{\sqrt{3}}{2}\)[/tex].
