Question

use reference angles to find the exact value of the following expression. do not use a calculator. sec 1380° select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. sec 1380° = (simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression. rationalize all denominators.) b. the answer is undefined.

Explanation:

Step1: Find the coterminal angle

To find a coterminal angle of \(1380^{\circ}\) between \(0^{\circ}\) and \(360^{\circ}\), divide \(1380\) by \(360\): \(1380\div360 = 3\) with a remainder. \(1380^{\circ}=3\times360^{\circ}+300^{\circ}\), so \(1380^{\circ}\) and \(300^{\circ}\) are coterminal.

Step2: Determine the reference angle

The angle \(300^{\circ}\) is in the fourth - quadrant. The reference angle \(\theta_{r}\) for an angle \(\theta = 300^{\circ}\) is \(360^{\circ}-300^{\circ}=60^{\circ}\).

Step3: Recall the secant function definition

We know that \(\sec\theta=\frac{1}{\cos\theta}\). In the fourth - quadrant, \(\cos\theta> 0\). And \(\cos300^{\circ}=\cos60^{\circ}=\frac{1}{2}\) (since the cosine of an angle and its reference angle in the fourth - quadrant have the same absolute value and the sign is positive in the fourth - quadrant).

Step4: Calculate the secant value

Since \(\sec\theta=\frac{1}{\cos\theta}\), and \(\cos300^{\circ}=\frac{1}{2}\), then \(\sec300^{\circ}=\frac{1}{\cos300^{\circ}} = 2\). So \(\sec1380^{\circ}=2\).

Answer:

A. \(\sec1380^{\circ}=2\)