Question

multiple choice 1 point give the exact value. sec 210° 2√3/3 -√2 -2√3/3 √2 2 multiple choice 1 point give the exact value. sec 240° -2√3/3 -2 2 2√3/3

Explanation:

Step1: Recall the definition of secant

$\sec\theta=\frac{1}{\cos\theta}$

Step2: Find the reference angle for \(210^{\circ}\)

The reference angle for \(210^{\circ}\) is \(210^{\circ}- 180^{\circ}=30^{\circ}\). And \(210^{\circ}\) is in the third - quadrant, where \(\cos\theta<0\). So \(\cos210^{\circ}=-\cos30^{\circ}=-\frac{\sqrt{3}}{2}\)

Step3: Calculate \(\sec210^{\circ}\)

Since \(\sec210^{\circ}=\frac{1}{\cos210^{\circ}}\), substituting \(\cos210^{\circ}=-\frac{\sqrt{3}}{2}\), we get \(\sec210^{\circ}=\frac{1}{-\frac{\sqrt{3}}{2}}=-\frac{2}{\sqrt{3}}=-\frac{2\sqrt{3}}{3}\)

Step4: Find the reference angle for \(240^{\circ}\)

The reference angle for \(240^{\circ}\) is \(240^{\circ}-180^{\circ} = 60^{\circ}\). And \(240^{\circ}\) is in the third - quadrant, where \(\cos\theta<0\). So \(\cos240^{\circ}=-\cos60^{\circ}=-\frac{1}{2}\)

Step5: Calculate \(\sec240^{\circ}\)

Since \(\sec240^{\circ}=\frac{1}{\cos240^{\circ}}\), substituting \(\cos240^{\circ}=-\frac{1}{2}\), we get \(\sec240^{\circ}=\frac{1}{-\frac{1}{2}}=- 2\)

Answer:

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