Sketch the curve represented by the parametric equations x=3cos¦È and y=3sin¦È for 0 ¡Ü ¦È ¡Ü ¦Ð?

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

Step 1: Determine the domain of θ. In this case, it is given that 0 ≤ θ ≤ π, which means we need to consider θ values from 0 to π.

Step 2: Choose a set of θ values within the given domain. We can start with some commonly used values such as 0, π/6, π/4, π/3, π/2, etc. You can choose as many points as you need to get a clear idea of the shape of the curve.

Step 3: Plug in the chosen values of θ into the x and y equations to find the corresponding x and y coordinates for each point.

For example, let's choose the following values of θ: 0, π/6, π/3, and π/2.

For θ = 0:
x = 3cos(0) = 3(1) = 3
y = 3sin(0) = 3(0) = 0
So, the point (3, 0) represents the curve at θ = 0.

For θ = π/6:
x = 3cos(π/6) = 3(√3/2) = (3√3)/2 ≈ 2.6
y = 3sin(π/6) = 3(1/2) = 3/2 ≈ 1.5
So, the point ((3√3)/2, 3/2) represents the curve at θ = π/6.

Similarly, we can calculate the coordinates for the other chosen values of θ:

For θ = π/3:
x = 3cos(π/3) = 3(1/2) = 3/2 ≈ 1.5
y = 3sin(π/3) = 3(√3/2) = (3√3)/2 ≈ 2.6
So, the point (3/2, (3√3)/2) represents the curve at θ = π/3.

For θ = π/2:
x = 3cos(π/2) = 3(0) = 0
y = 3sin(π/2) = 3(1) = 3
So, the point (0, 3) represents the curve at θ = π/2.

Step 4: Plot the calculated points on a coordinate plane. Once you have the coordinates for the chosen θ values, plot them on a graph. Connect the plotted points to form a smooth curve.

In this case, we have the following points:
(3, 0), ((3√3)/2, 3/2), (3/2, (3√3)/2), and (0, 3).

Plotting these points and connecting them gives us a quarter of a circle centered at the origin with radius 3.