To determine whether the point (\sqrt{#}, \sqrt{$}) lies on the unit circle, we use the equation of the unit circle:

$x^2 + y^2 = 1$

Step 1: Substitute the coordinates into the equation

Given the point (x, y) = (\sqrt{#}, \sqrt{$}):

(\sqrt{#})^2 + (\sqrt{$})^2 = 1

Step 2: Simplify the equation

# + $ = 1

If this condition is satisfied, then the point is on the unit circle.

Example 2:

For the point (\sqrt{#}, -\sqrt{$}) in Quadrant IV:

1. We know $y$ is negative in Quadrant IV, hence y = -\sqrt{$}.
2. To find its $y$-coordinate, we need to use the same unit circle equation:

(\sqrt{#})^2 + (-\sqrt{$})^2 = 1

This simplifies to:

# + $ = 1

Since the point is in Quadrant IV, you can directly compute its $y$-coordinate as -\sqrt{$}.

Conclusion:

1. Verify that # + $ = 1 for the first point to check if it lies on the unit circle.
2. The $y$-coordinate of the second point is -\sqrt{$} as expected in Quadrant IV.

Do you want more details or have any questions? Here are some related questions you might consider:

1. What values of # and $ will satisfy the equation for the unit circle?
2. How does the sign of $y$ determine the quadrant of a point?
3. What is the general form of the coordinates of points on the unit circle?
4. How can we visualize points in different quadrants on the unit circle?
5. What happens if the values of # and $ do not sum to 1?

Tip: Remember that the unit circle is defined by the radius being 1, which directly relates to the coordinates of points on it.