Question

give the exact value.
sec(210°)
a. √2
b. -√2
c. 2√3/3
d. -2√3/3

Explanation:

Step1: Use the reciprocal identity

Recall that \(\sec\theta=\frac{1}{\cos\theta}\). So, \(\sec(210^{\circ})=\frac{1}{\cos(210^{\circ})}\).

Step2: Find the reference angle

The angle \(210^{\circ}\) is in the third - quadrant. The reference angle \(\theta_{r}=210^{\circ}-180^{\circ} = 30^{\circ}\). And in the third - quadrant, \(\cos\theta<0\).

Step3: Evaluate \(\cos(210^{\circ})\)

We know that \(\cos(210^{\circ})=-\cos(30^{\circ})\). Since \(\cos(30^{\circ})=\frac{\sqrt{3}}{2}\), then \(\cos(210^{\circ})=-\frac{\sqrt{3}}{2}\).

Step4: Calculate \(\sec(210^{\circ})\)

Substitute \(\cos(210^{\circ}) =-\frac{\sqrt{3}}{2}\) into \(\sec(210^{\circ})=\frac{1}{\cos(210^{\circ})}\). We get \(\sec(210^{\circ})=\frac{1}{-\frac{\sqrt{3}}{2}}=-\frac{2}{\sqrt{3}}\). Rationalize the denominator: \(-\frac{2}{\sqrt{3}}\times\frac{\sqrt{3}}{\sqrt{3}}=-\frac{2\sqrt{3}}{3}\).

Answer:

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