What graph is traced by the parametric equation (x, y) = (2 + cos t, 1 + sin t)?

The parametric equation \((x(t), y(t)) = (2 + \cos t, 1 + \sin t)\) represents a **circle** in the plane. Let's analyze this step by step.

### Parametric Equations:
\[
x(t) = 2 + \cos t
\]
\[
y(t) = 1 + \sin t
\]

### Rewriting into a Standard Circle Form:
1. We know that \(\cos^2 t + \sin^2 t = 1\) for all \(t\).
2. Subtract 2 from \(x(t)\) and 1 from \(y(t)\):
   \[
   x - 2 = \cos t \quad \text{and} \quad y - 1 = \sin t
   \]
3. Now, square both sides and add them:
   \[
   (x - 2)^2 + (y - 1)^2 = \cos^2 t + \sin^2 t = 1
   \]
   
### Conclusion:
This is the equation of a circle with **center** at \((2, 1)\) and **radius** \(1\).

### Graphical Description:
The curve described by the parametric equation is a circle centered at \((2, 1)\) with radius \(1\).

Would you like further details or have any questions about this?

Here are 5 related questions to expand your understanding:
1. How would the graph change if the parametric equation was \((x, y) = (a + \cos t, b + \sin t)\)?
2. What would the graph look like if you changed \(\cos t\) to \(\cos(2t)\) in the parametric equation?
3. Can we use a different parametric representation for the same circle?
4. How can you generalize parametric equations to describe ellipses instead of circles?
5. How would you describe the curve if we replaced \(\cos t\) and \(\sin t\) with \(\tan t\) and \(\cot t\)?

**Tip**: Parametric equations are often used in computer graphics to model smooth curves such as circles and ellipses because they easily define continuous motion.

Explore the parametric equation (x, y) = (2 + cos t, 1 + sin t) that represents a circle with center at (2, 1) and radius 1. Learn how to rewrite parametric equations in standard form, using trigonometric identities. Suitable for high school students in Grades 10-12.

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