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 with \(1380^\circ\), we subtract multiples of \(360^\circ\) until we get an angle between \(0^\circ\) and \(360^\circ\).
We know that \(1380\div360 = 3\) with a remainder. So we calculate \(1380 - 3\times360=1380 - 1080 = 300^\circ\). So the coterminal angle of \(1380^\circ\) is \(300^\circ\).

Step2: Determine the reference angle

The angle \(300^\circ\) is in the fourth quadrant. The formula for the reference angle \(\theta'\) when the angle \(\theta\) is in the fourth quadrant is \(\theta'=360^\circ-\theta\). So for \(\theta = 300^\circ\), the reference angle \(\theta'=360 - 300=60^\circ\).

Step3: Recall the definition of secant and the sign in the fourth quadrant

We know that \(\sec\theta=\frac{1}{\cos\theta}\). In the fourth quadrant, cosine is positive, so secant is also positive.
We know that \(\cos60^\circ=\frac{1}{2}\), so \(\sec60^\circ=\frac{1}{\cos60^\circ} = 2\). Since \(\sec300^\circ=\sec(360^\circ - 60^\circ)=\sec60^\circ\) (because cosine is even and \(\cos(360^\circ-\alpha)=\cos\alpha\)) and \(\sec1380^\circ=\sec300^\circ\) (because they are coterminal), so \(\sec1380^\circ = 2\).

Answer:

\(\sec1380^{\circ}=\boldsymbol{2}\)