To solve this problem, let's understand how each transformation affects the terminal point on the unit circle.

Given:

• The terminal point at angle $t$ is $\left( \frac{8}{17}, \frac{15}{17} \right)$ on the unit circle.

For each of the transformations $\pi - t$, $-t$, $\pi + t$, and $2\pi + t$, we need to find the new coordinates of the terminal point.

Solution Steps:

1. (a) $\pi - t$:

   • Reflecting around $\pi$ means the point will be in the opposite quadrant on the unit circle.
   • This reflection changes the x-coordinate sign but keeps the y-coordinate the same.
   • So, the point becomes $\left( -\frac{8}{17}, \frac{15}{17} \right)$.

2. (b) $-t$:

   • Rotating to $-t$ reflects the point across the x-axis.
   • This reflection keeps the x-coordinate the same but changes the sign of the y-coordinate.
   • The point is $\left( \frac{8}{17}, -\frac{15}{17} \right)$.

3. (c) $\pi + t$:

   • Adding $\pi$ to $t$ reflects the point across the origin.
   • Both the x- and y-coordinates change signs.
   • The new point is $\left( -\frac{8}{17}, -\frac{15}{17} \right)$.

4. (d) $2\pi + t$:

   • Adding $2\pi$ to $t$ brings the point back to the original position, as it is a full rotation.
   • The coordinates remain the same: $\left( \frac{8}{17}, \frac{15}{17} \right)$.

Final Answers:

• (a) $(x, y) = \left( -\frac{8}{17}, \frac{15}{17} \right)$
• (b) $(x, y) = \left( \frac{8}{17}, -\frac{15}{17} \right)$
• (c) $(x, y) = \left( -\frac{8}{17}, -\frac{15}{17} \right)$
• (d) $(x, y) = \left( \frac{8}{17}, \frac{15}{17} \right)$

Would you like a further breakdown of any part of this solution?

Additional Questions:

1. What is the significance of each transformation in terms of the unit circle's quadrants?
2. How would the results change if the original point were in a different quadrant?
3. What is the geometric interpretation of each operation on $t$?
4. Can we generalize these transformations for any point $(x, y)$ on the unit circle?
5. How would this change if the problem involved an angle not on the unit circle?

Tip:

To understand angle transformations on the unit circle, remember that adding $\pi$ (180 degrees) reflects across the origin, while adding $2\pi$ (360 degrees) brings the point back to its original position.