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Parametric Equations Calculator

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Example
Created on 2024-06-20Asked by Amelia Ramirez (Solvelet student)
Find the Cartesian equation of the curve given by the parametric equations x=3cos⁡(t) x = 3\cos(t) and y=4sin⁡(t) y = 4\sin(t) .

Solution

To find the Cartesian equation of the curve given by the parametric equations x=3cos⁡(t) x = 3\cos(t) and y=4sin⁡(t) y = 4\sin(t) : 1. Express cos⁡(t) \cos(t) and sin⁡(t) \sin(t) in terms of x x and y y : cos⁡(t)=x3,sin⁡(t)=y4. \cos(t) = \frac{x}{3}, \quad \sin(t) = \frac{y}{4}. 2. Use the Pythagorean identity cos⁡2(t)+sin⁡2(t)=1 \cos^2(t) + \sin^2(t) = 1 : (x3)2+(y4)2=1, \left( \frac{x}{3} \right)^2 + \left( \frac{y}{4} \right)^2 = 1, x29+y216=1. \frac{x^2}{9} + \frac{y^2}{16} = 1. Therefore, the Cartesian equation of the curve is: x29+y216=1. \frac{x^2}{9} + \frac{y^2}{16} = 1. Solved on Solvelet with Basic AI Model
Some of the related questions asked by Mia Young on Solvelet
1. Eliminate the parameter t t from the parametric equations x=2t+1 x = 2t + 1 and y=t2−1 y = t^2 - 1 to obtain the Cartesian equation of the curve.2. Sketch the curve described by the parametric equations x=cos⁡(t)x = \cos(t) and y=sin⁡(t)y = \sin(t) for t∈[0,2π]t \in [0, 2\pi].
DefinitionA parametric equation is an equation that uses parameters to express the coordinates of the points that make up the curve. This form is useful for curves and surfaces. For example: x=cos(t), y=sin(t), t≤[0,2≤] (parameter) defines a circle with radius 1.
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