How can I eliminate the parameter to find the Cartesian equation of the curve defined by the expressions x = sin(12t) and y = cos(12t)?

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Finding the Cartesian Equation by Eliminating the Parameter

To convert the parametric equations into a Cartesian equation, we need to eliminate the parameter t from the given equations:

  • x = sin(12t)
  • y = cos(12t)

First, we can utilize the Pythagorean identity of sine and cosine:

sin²(θ) + cos²(θ) = 1

Applying this identity to our equations, we can say:

sin²(12t) + cos²(12t) = 1

Now substituting the x and y equations:

x² + y² = 1

This x² + y² = 1 is the Cartesian equation of the curve. It represents a circle centered at the origin with a radius of 1.

In conclusion, by eliminating the parameter t using trigonometric identities, we successfully derived the Cartesian equation from the given parametric equations.

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