Problem 51 - Precalculus

Question

Eliminate the parameter and obtain the standard form of the rectangular equation. Ellipse: \(x=h+a \cos \theta, y=k+b \sin \theta\)

Step-by-Step Solution

Answer

The rectangular form of the given ellipse equation is \((x - h)^2/a^2 + (y - k)^2/b^2 = 1\)

Step 1: Isolate the trigonometric functions

First, isolate \(\cos \theta\) and \(\sin \theta\) from the given equations. To do this, subtract \(h\) from the \(x\) equation to get: \(x-h = a \cos \theta\), then divide by \(a\) to isolate \(\cos \theta\) to get \(\cos \theta = (x - h)/a\).Do likewise to the \(y\) equation, subtract \(k\) to get \(y-k = b \sin \theta\), than divide by \(b\) to get \(\sin \theta = (y - k)/b\)

Step 2: Apply Pythagorean trigonometric identity

Then, replace \(\sin \theta\) and \(\cos \theta\) into the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\). This yields: \(((y - k)/b)^2 + ((x - h)/a)^2 = 1\)

Step 3: Simplify to obtain the rectangular equation

Upon simplifying, you receive the rectangular form of the ellipse equation: \((x - h)^2/a^2 + (y - k)^2/b^2 = 1\)

Key Concepts

Trigonometric Functions

Trigonometric functions are essential tools in mathematics, especially when working with shapes and angles. They describe relationships between the angles and sides of triangles, but they're also useful in modeling periodic phenomena. The primary trigonometric functions are sine (\sin) and cosine (\cos), which arise naturally from a right triangle.

• Sine (\sin): This function gives the ratio of the opposite side to the hypotenuse in a right triangle.
• Cosine (\cos): This function provides the ratio of the adjacent side to the hypotenuse.

In the context of ellipse equations, these functions help to describe the position of a point on the ellipse. By isolating \(\cos \theta\) and \(\sin \theta\), we can link the trigonometric parameters to rectangular coordinates.

Pythagorean Identity

The Pythagorean identity is a fundamental relationship in trigonometry, emerging from the Pythagorean theorem. It describes a critical property of sine and cosine functions:\[\sin^2 \theta + \cos^2 \theta = 1\]This equation states that the sum of the square of the sine and the square of the cosine of an angle always equals one. It is derived from the unit circle, where the radius is always 1. This identity is invaluable in manipulating and transforming equations involving trigonometric functions.
In our ellipse problem, substituting the expressions for \(\sin \theta\) and \(\cos \theta\) into the Pythagorean identity helps us to eliminate the parameter \(\theta\) and connect our parametric equations to a non-parametric, rectangular form.

Rectangular Equations

Rectangular equations express mathematical relationships directly in terms of \(x\) and \(y\) coordinates. They're often easier to interpret visually on a Cartesian plane. In this exercise, after isolating and substituting the trigonometric components, we converted from parametric to rectangular form.
The transformation gives us the standard form of the equation:\[(x - h)^2/a^2 + (y - k)^2/b^2 = 1\]This equation represents an ellipse in a familiar algebraic form, which is straightforward to graph and analyze. The terms \((x-h)\) and \((y-k)\) describe a translation of the ellipse’s center, and \(a\) and \(b\) represent the semi-major and semi-minor axes, respectively.

Parametric Equations

Parametric equations allow us to express geometric shapes more flexibly, using parameters to define the coordinates. For ellipses, this typically involves trigonometric functions. The given parametric equations are:

• \(x = h + a \cos \theta\)
• \(y = k + b \sin \theta\)

These equations represent an ellipse by sweeping a point around in a circular path defined by \(\theta\). The parameter \(\theta\) acts as an angle, dictating the point’s position.
By transitioning from parametric to rectangular equations, we remove \(\theta\) and focus solely on \(x\) and \(y\), making the equations more directly applicable for graphing or solving algebraically.