Final answer:
The trajectory of the object is an ellipse.
Step-by-step explanation:
The trajectory of the object can be determined by analyzing the given position function r(t). The position function has the form r(t) = asin(t)i + bcos(t)j, where i and j are unit vectors in the x and y directions, respectively. This equation represents motion in the xy-plane.
To determine the trajectory, we can rewrite the equation in terms of x and y coordinates. Using the trigonometric identities sin(t) = cos(t - π/2) and cos(t) = sin(t + π/2), we can rewrite the equation as r(t) = acos(t - π/2)i + bsin(t + π/2)j. This equation represents a rotation of the coordinate system by π/2 radians (90 degrees).
Since the x-coordinate is given by acos(t - π/2) and the y-coordinate is given by bsin(t + π/2), we can see that the equation represents an ellipse. Therefore, the trajectory of the object is an ellipse.