Question

if the sin 30° = \\(\frac{1}{2}\\), then which statement is true?

○ cos 150° = 0, because the angles are supplements

○ cos 60° = \\(\frac{1}{2}\\), because the angles are complements

○ cos 150° = 1, because the angles are supplements

○ cos 60° = 0, because the angles are complements

Explanation:

Step1: Recall Complementary Angles Property

Complementary angles sum to \(90^\circ\). For an angle \(\theta\), \(\sin\theta=\cos(90^\circ - \theta)\).

Step2: Check Angle Relationship

Given \(\theta = 30^\circ\), its complement is \(90^\circ-30^\circ = 60^\circ\). So \(\sin30^\circ=\cos60^\circ\).
Since \(\sin30^\circ=\frac{1}{2}\), then \(\cos60^\circ=\frac{1}{2}\) because \(30^\circ\) and \(60^\circ\) are complements.
Now check other options:

• For supplementary angles (sum \(180^\circ\)), \(\cos(180^\circ - \theta)=-\cos\theta\). \(180^\circ - 30^\circ = 150^\circ\), so \(\cos150^\circ=-\cos30^\circ=-\frac{\sqrt{3}}{2}

eq0,1\), so options with \(150^\circ\) are wrong.

• \(\cos60^\circ

eq0\) as shown, so the last option is wrong.

Answer:

\(\boldsymbol{\text{cos } 60^\circ = \frac{1}{2}, \text{ because the angles are complements}}\) (the second option)