https://file.aiquickdraw.com/mathbot/file/1725448164879pdmw6and.jpg

The problem presented involves the parametric equations:

\[
x(t) = \cos(t), \quad y(t) = \sin(t), \quad z(t) = 1 - \sin^2(t)
\]

The question asks about the shape of the space curve defined by these equations.

### Analyzing the Parametric Equations:
1. **\( x(t) = \cos(t) \)** and **\( y(t) = \sin(t) \)**: These are the standard parametric equations for a circle in the \(xy\)-plane with radius 1 centered at the origin \((0,0)\).
   
2. **\( z(t) = 1 - \sin^2(t) \)**: This can be rewritten using the Pythagorean identity, \( \sin^2(t) + \cos^2(t) = 1 \), as:
   \[
   z(t) = \cos^2(t)
   \]

Now, if we substitute the expressions for \(x(t)\) and \(z(t)\), we have:
\[
z = \cos^2(t) = x^2
\]

This implies that the curve described by these parametric equations is a parabola in the \(xz\)-plane, where \(z = x^2\). Since \(y(t) = \sin(t)\), this describes a circular motion in the \(xy\)-plane, meaning the curve forms a parabolic cylinder.

### Conclusion:
The shape of the space curve is **parabola**. Therefore, the correct answer is **(a) parabola**.

Would you like more details or have any further questions?

Here are 5 related questions that expand on this information:
1. How would the parametric equations change if the curve were an ellipse instead of a circle?
2. What is the geometric interpretation of a parabola in three-dimensional space?
3. Can you derive the equation of the curve if \( z(t) \) was given as \( z(t) = 1 - \cos^2(t) \)?
4. What would be the curve if the \( z(t) \) equation was \( z(t) = t \)?
5. How would the curve be affected if the parametric equations had a time-dependent phase shift?

**Tip:** When dealing with parametric equations, it is often helpful to eliminate the parameter to understand the overall shape of the curve in the coordinate system.

Consider the parametric equations x(t) = cos(t), y(t) = sin(t), z(t) = 1 - sin^2(t). What is the shape of the space curve defined by these equations?

Explore the shape of the space curve defined by the parametric equations x(t) = cos(t), y(t) = sin(t), and z(t) = 1 - sin^2(t). Understand how trigonometry and geometry come together to describe a parabola in the xz-plane. Suitable for Grades 10-12.

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