Question

Parametric equation for a circle. The parametric equation for a circle is x=rcos(∅) and y=r sin(∅) where r is the radius and ∅ is the angle of rotation counter-clockwise from the positive x axis. Defined this way, x and y satisfy the equation x² +y² =r² . Show this using OCTAVE. Use linspace to create an angle vector, theta, with values (0,π/3,2π/3,π,4π/3,5π/3,2π). Compute the corresponding x - and y-vectors for r=5. Show that the x - and y-vectors satisfy the equation of a circle.

Answer 1

Using OCTAVE, the parametric equations for a circle, x = r cos(φ) and y = r sin(φ), can be demonstrated by creating vectors for x and y using angles from theta and a given radius r. Validating the equation x² + y² = r² for each pair of x and y shows they form a circle of radius r.

Step-by-step explanation:

The parametric equations x = r cos(φ) and y = r sin(φ) describe a circle with radius r in the xy-plane. Using OCTAVE, we can show these equations represent a circle. First, we create a vector theta using linspace for angles 0 to 2π at the specified points. For a radius of 5, we can compute corresponding x and y vectors as follows:

• x = 5 * cos(theta)
• y = 5 * sin(theta)

To confirm these vectors lie on a circle, we check if they satisfy the equation x² + y² = r². To illustrate:

theta = linspace(0, 2*pi, 7); // Generates angles [0, π/3, 2π/3, π, 4π/3, 5π/3, 2π]
x = 5 * cos(theta); // Computes x-components
y = 5 * sin(theta); // Computes y-components
// Verify circle equation
display(x.^2 + y.^2) // Should all be 25 (r² since r=5)

When computing x² + y² for each theta, the result should be constant (25 in this case), confirming that the x and y vectors indeed form a circle of radius 5.