Question

Convert to a parametric equation.

a) x = sin(t), y = cos(t)
b) x = cos(t), y = sin(t)
c) x = 2t, y = t²
d) x = t², y = 2t

Answer 1

The question involves converting standard equations into parametric form. Parts (a) and (b) describe a circle, while parts (c) and (d) describe different parabolic trajectories, all using 't' as the parameter.

Step-by-step explanation:

The question involves converting given equations into their parametric forms. When equations are expressed in terms of a parameter (commonly t), each variable is defined as a function of that parameter.

a) The given equations x = sin(t) and y = cos(t) are already in parametric form, representing a circle with a radius of 1, centered at the origin in the Cartesian plane.

b) Similarly, x = cos(t) and y = sin(t) also represent a circle with a radius of 1, centered at the origin, parameterized differently from part (a).

c) For x = 2t and y = t2, these equations describe a parabolic trajectory, where t is the parameter. As t varies, x changes linearly, and y changes quadratically.

d) The final set, x = t2 and y = 2t also describes a parabolic path but oriented differently relative to the coordinate axes than in part (c).