To find the initial point on the circle equation $x^2 + y^2 = 16$ in the third quadrant, we need to understand a few key points:

1. The equation $x^2 + y^2 = 16$ represents a circle centered at the origin (0, 0) with a radius of 4.
2. In the third quadrant, both $x$ and $y$ are negative.
3. The initial point can be found by considering the standard parametric form of the circle $x = r \cos(\theta)$ and $y = r \sin(\theta)$, where $r = 4$ (the radius of the circle), and $\theta$ is the angle.

In the third quadrant:

• $x$ and $y$ are both negative, so we look for the angle $\theta$ that gives negative values for both.

In the third quadrant, the angle $\theta$ lies between $180^\circ$ and $270^\circ$, or between $\pi$ and $\frac{3\pi}{2}$ radians.

Let's use $\theta = 225^\circ$ (which is in the middle of the third quadrant).

Using parametric equations:

At $\theta = 225^\circ$:

• $x = 4 \cos(225^\circ) = 4 \times \left(-\frac{1}{\sqrt{2}}\right) = -\frac{4}{\sqrt{2}} = -2\sqrt{2}$
• $y = 4 \sin(225^\circ) = 4 \times \left(-\frac{1}{\sqrt{2}}\right) = -\frac{4}{\sqrt{2}} = -2\sqrt{2}$

Thus, the initial point on the circle in the third quadrant is $\left( -2\sqrt{2}, -2\sqrt{2} \right)$.

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Would you like further clarification or details on how to find points in other quadrants?

Here are five related questions:

1. How do parametric equations describe the points on a circle?
2. How do you calculate the coordinates of points in other quadrants of a circle?
3. What is the general form of the equation of a circle?
4. How can you find the center and radius of a circle from its equation?
5. How do you convert from degrees to radians?

Tip: When finding points in specific quadrants, remember that the signs of $x$ and $y$ change depending on the quadrant you're in!