https://r2.erweima.ai/mathbot/image/1725587197815yt9r13kt.jpg

The image contains the parametric equations for \(x\) and \(y\) as a function of a parameter \(t\), with a given domain:

\[
x = 4 \cos t, \quad y = 4 \sin t, \quad -\frac{\pi}{2} \leq t \leq \frac{\pi}{2}
\]

This represents a parametric equation for a segment of a circle with radius 4.

Let me break it down:

- The equations \(x = 4 \cos t\) and \(y = 4 \sin t\) describe a circle with a radius of 4.
- The range of \(t\) from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) indicates that this represents only part of the circle, specifically the right half of the circle (the first and fourth quadrants).

If you plot these parametric equations, you would see a semi-circle centered at the origin with a radius of 4.

Do you want further details on how to graph this or how parametric equations work? Here are some related questions:

1. How do we plot parametric equations step-by-step?
2. What is the geometric interpretation of \(\cos t\) and \(\sin t\) in these equations?
3. How does restricting \(t\) to \([- \frac{\pi}{2}, \frac{\pi}{2}]\) affect the graph?
4. How do we convert parametric equations to a single Cartesian equation?
5. What are the typical applications of parametric equations in real-life problems?

**Tip:** When dealing with parametric equations, always check the range of the parameter to understand which portion of the graph is represented.

x = 4 cos t, y = 4 sin t, -π/2 ≤ t ≤ π/2

Explore the parametric equations x = 4 cos t and y = 4 sin t with the restriction -π/2 ≤ t ≤ π/2, representing a semi-circle with a radius of 4. This problem covers key trigonometric concepts and the geometric representation of parametric equations, ideal for students in Grades 10-12.

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