How do you know what shape the graph in the xy-plane represented by x=3sin(t) and y=2cos(t) is?

x/3=sin(t), y/2=cos(t)

x^2/3^2=sin^2(t)
+
y^2/2^2=cos^2(t)
_____________________

x^2/3^2+y^2/2^2=1

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To determine the shape of the graph represented by the equations x = 3sin(t) and y = 2cos(t), you can follow these steps:

1. Understand the given equations:
- x = 3sin(t): The value of x is determined by the sine function of t, multiplied by 3.
- y = 2cos(t): The value of y is determined by the cosine function of t, multiplied by 2.

2. Analyze the trigonometric functions:
- The sine function (sin(t)) oscillates between -1 and 1. When multiplied by a constant (in this case, 3), it stretches or shrinks the range.
- The cosine function (cos(t)) also oscillates between -1 and 1. When multiplied by a constant (2 in this case), it modifies the amplitude.

3. Understand the parameter t:
- The parameter t represents an angle. As it varies, the values of x and y will change accordingly.

4. Determine the shape:
- Since x is determined by 3sin(t) and y is determined by 2cos(t), the graph will take the form of a curve in the xy-plane.
- This curve is known as an ellipse. When you plot all the points (x, y) on the graph, they will form an elliptical shape.

By following these steps, you can conclude that the graph represented by x = 3sin(t) and y = 2cos(t) is an ellipse in the xy-plane.