To determine which statement is true, let’s use the given information and trigonometric identities.
Given:
[tex]\[
\sin 30^\circ = \frac{1}{2}
\][/tex]
We need to determine the true statement among the following options:
1. [tex]\(\cos 150^\circ = 0\)[/tex], because the angles are supplements
2. [tex]\(\cos 60^\circ = \frac{1}{2}\)[/tex], because the angles are complements
3. [tex]\(\cos 150^\circ = 1\)[/tex], because the angles are supplements
4. [tex]\(\cos 60^\circ = 0\)[/tex], because the angles are complements
Step-by-step solution:
1. Complementary Angles Identity:
Complementary angles are two angles whose sum is [tex]\(90^\circ\)[/tex]. For complementary angles [tex]\(\theta\)[/tex] and [tex]\(90^\circ - \theta\)[/tex], the trigonometric identity is:
[tex]\[
\cos(90^\circ - \theta) = \sin(\theta)
\][/tex]
Here, if we apply this identity with [tex]\(\theta = 30^\circ\)[/tex]:
[tex]\[
\cos(90^\circ - 30^\circ) = \cos 60^\circ = \sin 30^\circ
\][/tex]
Given [tex]\(\sin 30^\circ = \frac{1}{2}\)[/tex], we have:
[tex]\[
\cos 60^\circ = \frac{1}{2}
\][/tex]
Therefore, the statement: [tex]\(\cos 60^\circ = \frac{1}{2}\)[/tex], because the angles are complements, is true.
2. Supplementary Angles Identity:
Supplementary angles are two angles whose sum is [tex]\(180^\circ\)[/tex]. For supplementary angles [tex]\(\theta\)[/tex] and [tex]\(180^\circ - \theta\)[/tex], the trigonometric identity is:
[tex]\[
\cos(180^\circ - \theta) = -\cos(\theta)
\][/tex]
Considering [tex]\(\theta = 30^\circ\)[/tex]:
[tex]\[
\cos 150^\circ = \cos(180^\circ - 30^\circ) = -\cos 30^\circ
\][/tex]
[tex]\[
\cos 30^\circ = \frac{\sqrt{3}}{2}
\][/tex]
[tex]\[
\cos 150^\circ = -\frac{\sqrt{3}}{2}
\][/tex]
Therefore, none of the statements about [tex]\(\cos 150^\circ\)[/tex] being 0 or 1 are correct.
Conclusively, the only statement that holds true is:
[tex]\[
\cos 60^\circ = \frac{1}{2}, \text{ because the angles are complements}
\][/tex]
