Question

use reference angles to find the exact value of the following expression. do not use a calculator. sec 930° select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. sec 930° = (simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression. r b. the answer is undefined.

Explanation:

Step1: Reduce the angle by 360° multiples

To find the coterminal angle of \(930^\circ\), we subtract multiples of \(360^\circ\).
\(930^\circ - 2\times360^\circ = 930^\circ - 720^\circ = 210^\circ\)
So, \(\sec 930^\circ=\sec 210^\circ\)

Step2: Find the reference angle

The angle \(210^\circ\) is in the third quadrant. The reference angle \(\theta_{ref}\) for an angle \(\theta\) in the third quadrant is \(\theta - 180^\circ\).
So, \(\theta_{ref}=210^\circ - 180^\circ = 30^\circ\)

Step3: Determine the sign of secant

Secant is the reciprocal of cosine. In the third quadrant, cosine (and thus secant) is negative. So, \(\sec 210^\circ=-\sec 30^\circ\)

Step4: Recall the value of \(\sec 30^\circ\)

We know that \(\cos 30^\circ=\frac{\sqrt{3}}{2}\), so \(\sec 30^\circ = \frac{1}{\cos 30^\circ}=\frac{2}{\sqrt{3}}=\frac{2\sqrt{3}}{3}\) (rationalizing the denominator)

Step5: Find the value of \(\sec 210^\circ\)

Since \(\sec 210^\circ=-\sec 30^\circ\), we substitute the value of \(\sec 30^\circ\)
\(\sec 210^\circ=-\frac{2\sqrt{3}}{3}\)

Answer:

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