Question

x=2 sin (t), y=2 cos(t),0 ≤ t ≤ π A pair of parametric equations is given. Find a rectangular-coordinate equation for the curve by eliminating the parameter.

Answer 1

To find a rectangular-coordinate equation for a given pair of parametric equations, we can eliminate the parameter by using trigonometric identities. By expressing sine and cosine in terms of x and y, we can use the Pythagorean identity to simplify the equation and obtain a rectangular-coordinate equation.

Step-by-step explanation:

To eliminate the parameter and find a rectangular-coordinate equation for the given pair of parametric equations, we can use the trigonometric identities for sine and cosine.

From the given equations, x = 2sin(t) and y = 2cos(t), we can express sin(t) and cos(t) in terms of x and y:

sin(t) = x/2 and cos(t) = y/2.

Squaring both equations and adding them together, we get:

(sin(t))^2 + (cos(t))^2 = (x/2)^2 + (y/2)^2

Using the identity sin^2(t) + cos^2(t) = 1, we simplify the equation to:

1 = (x^2 + y^2)/4.

Multiplying both sides by 4, we get the rectangular-coordinate equation:

4 = x^2 + y^2.