We are given that the terminal point determined by $t$ on the unit circle is $\left( \frac{3}{5}, \frac{4}{5} \right)$. We are asked to find the terminal points for the following values of $t$:

Part (a) $\pi - t$

On the unit circle, subtracting $t$ from $\pi$ reflects the point across the y-axis. This means the x-coordinate changes sign, while the y-coordinate remains the same.

So the terminal point for $\pi - t$ is: $\left( -\frac{3}{5}, \frac{4}{5} \right)$

Part (b) $-t$

On the unit circle, changing $t$ to $-t$ reflects the point across the x-axis. This means the y-coordinate changes sign, while the x-coordinate remains the same.

So the terminal point for $-t$ is: $\left( \frac{3}{5}, -\frac{4}{5} \right)$

Summary of Terminal Points

• $\pi - t$ gives $\left( -\frac{3}{5}, \frac{4}{5} \right)$
• $-t$ gives $\left( \frac{3}{5}, -\frac{4}{5} \right)$

Would you like any further clarifications or additional details?

Here are some related questions you might consider:

1. What happens to the coordinates when reflecting across the origin (i.e., for $\pi + t$)?
2. How can we generalize the effect of adding or subtracting angles from $t$ in different quadrants of the unit circle?
3. What are the coordinates of the terminal point when $t = \frac{\pi}{2}$ or $t = 2\pi$?
4. Can you describe the relationship between sine and cosine when shifting angles by $\pi$ or by multiples of $\frac{\pi}{2}$?
5. How does the Pythagorean identity $\sin^2(t) + \cos^2(t) = 1$ relate to terminal points on the unit circle?

Tip: On the unit circle, transformations involving $\pi$ (like $\pi - t$ or $t + \pi$) always correspond to flipping signs of coordinates (reflections) depending on the specific axis.