If [tex]\sin 30^{\circ} = \frac{1}{2}[/tex], then which statement is true?

A. [tex]\cos 60^{\circ} = \frac{1}{2}[/tex], because the cosine and sine are complements.
B. [tex]\cos 60^{\circ} = 0[/tex], because the cosine and sine are complements.
C. [tex]\cos 150^{\circ} = 0[/tex], because the cosine and sine are supplements.
D. [tex]\cos 150^{\circ} = 1[/tex], because the cosine and sine are supplements.



Answer :

To determine which statement is true, follow these steps:

1. Understanding Trigonometric Complements:
- The sine and cosine of complementary angles (angles that add up to 90 degrees) exhibit a specific relationship: [tex]\(\sin x = \cos (90^\circ - x)\)[/tex].

2. Given Information:
- We know that [tex]\(\sin 30^\circ = \frac{1}{2}\)[/tex].

3. Finding [tex]\(\cos 60^\circ\)[/tex]:
- Since 60° and 30° are complementary angles ([tex]\(30^\circ + 60^\circ = 90^\circ\)[/tex]), we can use the complementary angles relationship.
- Thus, [tex]\(\cos 60^\circ\)[/tex] is the same as [tex]\(\sin 30^\circ\)[/tex].

4. Conclusion:
- Given [tex]\(\sin 30^\circ = \frac{1}{2}\)[/tex], it follows that [tex]\(\cos 60^\circ = \frac{1}{2}\)[/tex].

5. Verification of Statements:
- Therefore, the statement "[tex]\(\cos 60^\circ = \frac{1}{2}\)[/tex], because the cosine and sine are complements" is true.

Examining the incorrect options:
- [tex]\(\cos 60^\circ = 0\)[/tex]: This is incorrect. [tex]\(\cos 60^\circ = \frac{1}{2}\)[/tex].
- [tex]\(\cos 150^\circ = 0\)[/tex]: This is incorrect. The cosine of 150° is not 0.
- [tex]\(\cos 150^\circ = 1\)[/tex]: This is incorrect. The cosine of 150° is not 1.

Hence, the correct statement is:

[tex]\[ \cos 60^\circ = \frac{1}{2}, \text{ because the cosine and sine are complements}. \][/tex]