The diagram shows a semicircle - HSC - SSCE Mathematics Extension 1 - Question 9 - 2021 - Paper 1

To determine the correct parametric equations representing the semicircle, we need to focus on the center and radius of the semicircle as shown in the diagram.

Analyzing the Semicircle

• The semicircle is centered at (3, 2) with a radius of 1.
• The semicircle is oriented such that it is on the top half of the Cartesian plane.

Evaluating the Options

1. Option A:

   • $x = 3 + \sin t$ and $y = 2 + \cos t$
   • This represents a semicircle centered at (3, 2) above the center, but it needs $t$ from $\frac{\pi}{2}$ to $\pi$. This is a closed arc downward, which does not represent this semicircle.

2. Option B:

   • $x = 3 + \cos t$ and $y = 2 + \sin t$
   • This represents the upper semicircle centered at (3, 2) as $t$ varies from $0$ to $\pi$. So $t$ from $\frac{\pi}{2}$ to $\pi$ is consistent with the rest of the x-values.

3. Option C:

   • $x = 3 - \sin t$ and $y = 2 + \cos t$
   • This represents a semicircle but in the wrong orientation, descending.

4. Option D:

   • $x = 3 - \cos t$ and $y = 2 - \sin t$
   • This also represents a semicircle but is inappropriately positioned and inverted.

Conclusion

• Based on the analysis, Option B is the only choice that accurately reflects the parameters and orientation of the semicircle given.