To solve this problem, we need to find the terminal point on the unit circle for the angle $\frac{7\pi}{4}$ radians. Here's how we approach it:

1. Understand the angle position:

   • The angle $\frac{7\pi}{4}$ radians is in the fourth quadrant because it is greater than $\frac{3\pi}{2}$ radians but less than $2\pi$ radians.

2. Find the reference angle:

   • The reference angle is the acute angle formed with the x-axis. For $\frac{7\pi}{4}$, the reference angle is: $2\pi - \frac{7\pi}{4} = \frac{\pi}{4}.$

3. Determine the coordinates:

   • The unit circle coordinates for an angle with reference angle $\frac{\pi}{4}$ are: $(\cos(\frac{\pi}{4}), \sin(\frac{\pi}{4})) = \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right).$
   • In the fourth quadrant, the cosine (x-coordinate) is positive, and the sine (y-coordinate) is negative. Thus, the coordinates are: $\left( \frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2} \right).$

Final Answer:

The terminal point on the unit circle is: $\boxed{\left( \frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2} \right)}.$

Let me know if you need further details or explanations!

Related Questions:

1. What is the terminal point for $\frac{5\pi}{4}$ radians on the unit circle?
2. How do you find the terminal point for negative angles?
3. What are the coordinates of the terminal point for $\frac{\pi}{3}$ radians?
4. Why is the reference angle important for finding terminal points?
5. How do you use symmetry of the unit circle to determine terminal points quickly?

Tip:

Always determine the quadrant of the angle first. This will help you correctly assign signs to the coordinates.