The problem asks for the exact value of the cosine function, $\cos \theta$, where the angle $\theta$ is given as $\frac{7\pi}{4}$.

Steps to Solve:

1. Determine the Angle's Quadrant: $\frac{7\pi}{4}$ is in the fourth quadrant because: $\frac{7\pi}{4} \text{ radians} = 360^\circ - 45^\circ = 315^\circ$ Angles between $270^\circ$ and $360^\circ$ lie in the fourth quadrant.

2. Cosine Function in the Fourth Quadrant: The cosine function is positive in the fourth quadrant.

3. Reference Angle: The reference angle for $\frac{7\pi}{4}$ is $\frac{\pi}{4}$ or $45^\circ$.

4. Evaluate $\cos \theta$: $\cos \left(\frac{7\pi}{4}\right) = \cos \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$

Answer:

The correct value is B) $\frac{\sqrt{2}}{2}$.

Would you like any further details or have any questions?

Here are 8 related questions:

1. What are the values of sine and tangent for $\theta = \frac{7\pi}{4}$?
2. Can you explain the cosine function's behavior in each quadrant?
3. How do reference angles help in finding trigonometric function values?
4. What is the significance of the unit circle in trigonometry?
5. What are the trigonometric identities involving cosine?
6. How do you convert from degrees to radians and vice versa?
7. What is the importance of the CAST rule in trigonometry?
8. How do the sine and cosine functions relate to the Pythagorean identity?

Tip: When working with trigonometric functions, always verify which quadrant the angle lies in, as this determines the sign of the function's value.