Question

Eliminate the parameter in the given parametric equations. \(x=4 \sin t+1, \quad y=3 \cos t-2\)

Short answer

The Cartesian equation of the curve is \(\frac{(x-1)^2}{16} + \frac{(y+2)^2}{9} = 1\).

Step-by-step solution

Solve for Sine and Cosine

We start by expressing sine and cosine terms from the given parametric equations. From the equation for \(x\), solve for \(\sin t\): \(\sin t = \frac{x-1}{4}\). From the equation for \(y\), solve for \(\cos t\): \(\cos t = \frac{y+2}{3}\).

Apply the Pythagorean Identity

The Pythagorean identity tells us that \(\sin^2 t + \cos^2 t = 1\). Substitute the expressions from Step 1 into this identity: \(\left(\frac{x-1}{4}\right)^2 + \left(\frac{y+2}{3}\right)^2 = 1\).

Simplify the Expression

Expand and simplify the equation from Step 2: \(\frac{(x-1)^2}{16} + \frac{(y+2)^2}{9} = 1\). This equation represents an ellipse in the Cartesian coordinate plane.

Key concepts

Sine and Cosine

In trigonometry, sine and cosine are fundamental functions that are pivotal in describing oscillations, waves, and circular motion. These functions are defined based on a unit circle, which is a circle with a radius of 1, centered at the origin of a coordinate plane.
The two functions, sine (\( ext{sin}\)) and cosine (\( ext{cos}\)), describe the coordinates of a point on this unit circle. Specifically, for an angle \(t\), measured from the positive x-axis, the cosine of \(t\) gives the x-coordinate, while the sine of \(t\) gives the y-coordinate.
When working with parametric equations such as \(x=4 ext{sin} t+1\), the expression dictates how the x-coordinate changes with the angle. Similarly, \(y=3 ext{cos} t-2\) governs the y-coordinate's variation with the angle.
By extracting sine and cosine values from the parametric equations, we can analyze and link them to real-world applications such as physics, engineering, and computer graphics where periodic behavior is significant.

Pythagorean Identity

The Pythagorean Identity is a fundamental theorem in trigonometry. It states that for any angle \(t\), the square of sine plus the square of cosine equals one: \( ext{sin}^2 t + ext{cos}^2 t = 1\). This identity underscores the intrinsic relationship between the sine and cosine functions.
In the context of parametric equations, applying the Pythagorean identity allows us to transition from trigonometric functions to Cartesian coordinates. This is crucial when we aim to eliminate the parameter \(t\) and convert the equations into a form that is easier to handle.
For the given equations, substituting \( ext{sin} t = \frac{x-1}{4}\) and \( ext{cos} t = \frac{y+2}{3}\) into the identity provides a pathway to simplify the equations and helps derive the relationship between \(x\) and \(y\). This step is essential in revealing the geometric nature of the equation, which, in this case, leads us to the geometry of an ellipse.

Ellipse Equation

An ellipse is a fascinating shape characterized by its elongated circular form. In mathematics, an ellipse is defined by its equation in the Cartesian plane. The general form of the equation of an ellipse is: \\[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\] \ where \((h,k)\) is the center of the ellipse, and \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes, respectively.
By simplifying the parametric equations using the Pythagorean Identity, we derived: \(\frac{(x-1)^2}{16} + \frac{(y+2)^2}{9} = 1\). This translates into an ellipse with a center at \((1, -2)\), a semi-major axis of length 4, and a semi-minor axis of length 3.
Understanding these parameters is crucial because they determine the size, shape, and position of the ellipse in the plane. Ellipses are ubiquitous in physics, appearing in planetary orbits and optics, where they reflect light uniquely.