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]
