Question

1. find all angles θ in the interval 0,2π) that satisfy the expression secθ = -√2.

Explanation:

Step1: Recall the reciprocal identity

Since \(\sec\theta=\frac{1}{\cos\theta}\), the equation \(\sec\theta =-\frac{\sqrt{2}}{1}\) becomes \(\frac{1}{\cos\theta}=-\sqrt{2}\), then \(\cos\theta=-\frac{1}{\sqrt{2}}=-\frac{\sqrt{2}}{2}\).

Step2: Find the reference angle

We know that \(\cos\alpha=\frac{\sqrt{2}}{2}\) when \(\alpha = \frac{\pi}{4}\).

Step3: Determine the quadrants

Since \(\cos\theta<0\), \(\theta\) is in the second or third quadrant.
In the second quadrant, \(\theta=\pi-\frac{\pi}{4}=\frac{3\pi}{4}\).
In the third quadrant, \(\theta=\pi+\frac{\pi}{4}=\frac{5\pi}{4}\).

Answer:

\(\theta=\frac{3\pi}{4},\frac{5\pi}{4}\)