19. Sketch the curve represented by the parametric equations x = 3 cos θ and y = 3 sin θ for 0≤θ≤π

visit wolframalpha.com and enter

plot x=3cosθ,y=3sinθ

note that x^2+y^2 = 9

To sketch the curve represented by the parametric equations x = 3 cos(θ) and y = 3 sin(θ) for 0 ≤ θ ≤ π, follow these steps:

1. Start by plotting the initial point at θ = 0.
- Plug in θ = 0 into the parametric equations:
x = 3 cos(0) = 3
y = 3 sin(0) = 0
- So the initial point is (3, 0).

2. Determine the ending point at θ = π.
- Plug in θ = π into the parametric equations:
x = 3 cos(π) = -3
y = 3 sin(π) = 0
- So the ending point is (-3, 0).

3. Find other points along the curve by choosing values of θ between 0 and π and substituting into the parametric equations.
- For example, you can choose θ = π/4:
x = 3 cos(π/4) = 3 * (√2 / 2) = 3√2/2
y = 3 sin(π/4) = 3 * (√2 / 2) = 3√2/2
So the point is (3√2/2, 3√2/2).
- Similarly, you can choose other values of θ and find corresponding points.

4. Connect the points you found to sketch the curve.
- Join the plotted points on the graph with a smooth curve.
- The curve should start at (3, 0), pass through points like (3√2/2, 3√2/2), and end at (-3, 0).

By following these steps, you'll be able to sketch the curve represented by the given parametric equations.