Question

2 multiple choice 1 point
give the exact value.
$\cos 210^{\circ}$
$\frac{\sqrt{3}}{2}$
$-\frac{\sqrt{2}}{2}$
$\frac{\sqrt{2}}{2}$
$-\frac{\sqrt{3}}{2}$

Explanation:

Step1: Determine the reference angle

Since \(210^{\circ}=180^{\circ} + 30^{\circ}\), the reference angle \(\theta'=210^{\circ}-180^{\circ}=30^{\circ}\)

Step2: Determine the sign of the cosine function

The angle \(210^{\circ}\) is in the third - quadrant. In the third - quadrant, \(\cos\theta=\frac{x}{r}\) (where \(x<0\) and \(r > 0\)), so \(\cos210^{\circ}<0\)

Step3: Use the cosine of the reference angle

We know that \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\). Using the formula \(\cos(180^{\circ}+\alpha)=-\cos\alpha\) (here \(\alpha = 30^{\circ}\)), we have \(\cos210^{\circ}=-\cos30^{\circ}\)

Answer:

\(-\frac{\sqrt{3}}{2}\)