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

Trigonometry · Axiom Academy

EXAMPLE Graphing Parametric Equations Learn to eliminate the parameter and identify curves from parametric form Excellent work! You've successfully eliminated the parameter to identify an ellipse. Here's what we learned: Parametric equations with sine and cosine: When you see x = A·cos(t) and y = B·sin(t), expect a circle (if A = B) or an ellipse (if A ≠ B) Use the Pythagorean identity: The identity sin²(t) + cos²(t) = 1 is your key tool for eliminating the parameter t Isolate the trig functions: Solve for cos(t) and sin(t) separately, then substitute into the identity Standard form reveals properties: The equation x²/a² + y²/b² = 1 tells us the semi-axis lengths: here a = 2 (x-direction) and b = 3 (y-direction) Identify the major axis: The larger denominator indicates the major axis direction; in our case, 9 > 4, so the ellipse is taller than it is wide This method works for any parametric equations involving sine and cosine. Practice with different coefficients to build confidence identifying various ellipses!

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