Question 6 (Multiple Choice)

If [tex]\cos 30^{\circ} = \frac{\sqrt{3}}{2}[/tex], then [tex]\sin 60^{\circ} = [/tex]

A. 0, because the angles are complementary
B. [tex]\frac{1}{2}[/tex]
C. [tex]\frac{\sqrt{3}}{2}[/tex], because the angles are complementary
D. 1, because the angles are complementary



Answer :

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.