Obtain all the values theta between 0 and 2pi such that = (1)/(sqrt(2)).

Question

ScanSolve AI Tutor · Accepted Answer

Here are the solutions to the problems:

Question 72: Obtain all the values $\theta$ between 0 and $2\pi$ such that $\cos\theta = \frac{1}{\sqrt{2}}$ .

Find the principal value of $\theta$ .
We know that $\cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}$ . So, one value is $\theta_1 = \frac{\pi}{4}$ .
Find other values in the interval $[0, 2\pi]$ .
The cosine function is positive in the first and fourth quadrants.
The angle in the first quadrant is $\frac{\pi}{4}$ .
The angle in the fourth quadrant is $2\pi - \frac{\pi}{4} = \frac{8\pi - \pi}{4} = \frac{7\pi}{4}$ .
Both $\frac{\pi}{4}$ and $\frac{7\pi}{4}$ are within the interval $0 \le \theta \le 2\pi$ .

The final answer is $\boxed{\text{B. } \frac{\pi}{4} \text{ and } \frac{7\pi}{4}}$ .

**

Step-by-step solution