Question

which of the following angle(s) in standard position (in radians), if any, is $sec(\theta)=-\frac{\sqrt{2}}{2}$?
$\frac{3\pi}{4}\\&\frac{5\pi}{4}$
no such angle(s) in standard position exist.
$\frac{\pi}{4}\\&\frac{7\pi}{4}$
$\frac{\pi}{4}\\&\frac{3\pi}{4}$

Explanation:

Step1: Recall the secant - cosine relationship

We know that \(\sec\theta=\frac{1}{\cos\theta}\). Given \(\sec\theta =-\frac{\sqrt{2}}{2}\), then \(\cos\theta=\frac{1}{\sec\theta}=-\sqrt{2}\).

Step2: Analyze the range of cosine function

The range of the cosine function is \([- 1,1]\). Since \(-\sqrt{2}\approx - 1.414<-1\), there is no real - valued angle \(\theta\) (in standard position) for which \(\cos\theta =-\sqrt{2}\).

Answer:

No such angle(s) in standard position exist.