Question

compute sec 150°. the movement of the progress bar may be uneven because questions can be worth more or less (including zero) depending on your answer. options: 2, \\(\frac{2\sqrt{3}}{3}\\), \\(-\frac{2\sqrt{3}}{3}\\), \\(-\frac{\sqrt{3}}{2}\\)

Explanation:

Step1: Use the reciprocal identity of secant

We know that \(\sec\theta=\frac{1}{\cos\theta}\). So, \(\sec150^{\circ}=\frac{1}{\cos150^{\circ}}\).

Step2: Find the value of \(\cos150^{\circ}\)

\(150^{\circ}=180^{\circ} - 30^{\circ}\). Using the formula \(\cos(A - B)=\cos A\cos B+\sin A\sin B\) (here \(A = 180^{\circ}\), \(B=30^{\circ}\)), \(\cos180^{\circ}=- 1\), \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\), \(\sin180^{\circ}=0\), \(\sin30^{\circ}=\frac{1}{2}\). Then \(\cos150^{\circ}=\cos(180^{\circ}-30^{\circ})=\cos180^{\circ}\cos30^{\circ}+\sin180^{\circ}\sin30^{\circ}=(-1)\times\frac{\sqrt{3}}{2}+0\times\frac{1}{2}=-\frac{\sqrt{3}}{2}\).

Step3: Calculate \(\sec150^{\circ}\)

Since \(\sec150^{\circ}=\frac{1}{\cos150^{\circ}}\), substituting \(\cos150^{\circ}=-\frac{\sqrt{3}}{2}\), we get \(\sec150^{\circ}=\frac{1}{-\frac{\sqrt{3}}{2}}=-\frac{2\sqrt{3}}{3}\).

Answer:

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