Question

Convert parametric equations to Cartesian using a calculator.

A) x = cos(t), y = sin(t)
B) x = 2t, y = t²
C) x = t², y = 2t
D) x = 3sin(t), y = 3cos(t)

Answer 1

To convert parametric equations to Cartesian equations, we can eliminate the parameter and express the equations in terms of x and y. Each set of given equations can be converted to a corresponding Cartesian equation: a circle, a parabola, a straight line, or an ellipse.

So, the correct answer is option D.

Step-by-step explanation:

To convert parametric equations to Cartesian equations, we need to eliminate the parameter and express the equations in terms of x and y. Let's go through each given set of equations:

A) x = cos(t), y = sin(t)

Since x = cos(t) and y = sin(t), we can square both equations and use the trigonometric identity cos²(t) + sin²(t) = 1 to get x² + y² = 1, which is the Cartesian equation of a circle with radius 1 centered at the origin.

B) x = 2t, y = t²

By substituting x = 2t into y = t², we get y = (x/2)² or 4y = x². This is a Cartesian equation of a parabola.

C) x = t², y = 2t

By substituting x = t² into y = 2t, we get y = 2√x. This is a Cartesian equation of a straight line.

D) x = 3sin(t), y = 3cos(t)

Using the trigonometric identity sin²(t) + cos²(t) = 1, we can rewrite the equations as x²/9 + y²/9 = 1, which is the Cartesian equation of an ellipse.