The image asks to find the values of $t$ corresponding to given terminal points on the unit circle. The terminal points are expressed as pairs $P(x, y)$, and the parameter $t$ represents the angle in radians from the positive $x$-axis to the line segment connecting the origin to the point $P(x, y)$.

Let's analyze and solve for $t$ for each terminal point.

(a) $P(0, 1)$

This point lies directly on the positive $y$-axis, which corresponds to an angle of: $t = \frac{\pi}{2}$

(b) $P\left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$

This point is in the second quadrant, and it represents a standard angle of $\frac{3\pi}{4}$.

(c) $P(-1, 0)$

This point lies on the negative $x$-axis, corresponding to: $t = \pi$

(d) $P\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)$

This point is in the third quadrant, corresponding to: $t = \frac{5\pi}{4}$

(e) $P(0, -1)$

This point lies on the negative $y$-axis, corresponding to: $t = \frac{3\pi}{2}$

(f) $P\left(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)$

This point is in the fourth quadrant, corresponding to: $t = \frac{7\pi}{4}$

Summary of $t$-values:

• (a) $t = \frac{\pi}{2}$
• (b) $t = \frac{3\pi}{4}$
• (c) $t = \pi$
• (d) $t = \frac{5\pi}{4}$
• (e) $t = \frac{3\pi}{2}$
• (f) $t = \frac{7\pi}{4}$

Would you like more details on how to find these values or have any further questions?

Here are 5 related questions to think about:

1. What is the general form for the parametric equation of the unit circle?
2. How do you determine the quadrant based on the coordinates $(x, y)$?
3. What is the relationship between angles in radians and degrees?
4. Why are the angles in the unit circle symmetric across the origin?
5. How can you calculate the coordinates for any given angle $t$?

Tip: When working with the unit circle, memorize key points like $\frac{\pi}{4}$, $\frac{\pi}{2}$, and $\pi$, as these correspond to commonly used angles with simple coordinates!