Problem 17 - Precalculus with Calculus Previews

Question

Eliminate the parameter from the given set of parametric equations and obtain a rectangular equation that has the same graph. $$ x=4 \cos t, y=2 \sin t, 0 \leq t \leq 2 \pi $$

Step-by-Step Solution

Answer

The rectangular equation is \(x^2 + 4y^2 = 16\), representing an ellipse.

Step 1: Express x and y in terms of trigonometric identities

The given parametric equations are: \[ x = 4 \cos t \quad \text{and} \quad y = 2 \sin t \]Notice these equations resemble the parametric representation of an ellipse. We need to rewrite these equations using trigonometric identities.

Step 2: Isolate cosine and sine

Divide the equation for \(x\) and \(y\) by their respective coefficients to express \(\cos t\) and \(\sin t\):\[ \cos t = \frac{x}{4} \quad \text{and} \quad \sin t = \frac{y}{2} \]

Step 3: Apply the Pythagorean identity

Use the Pythagorean identity \( \cos^2 t + \sin^2 t = 1 \) to eliminate the parameter \(t\):Substitute \( \cos t \) and \( \sin t \) into the identity:\[ \left( \frac{x}{4} \right)^2 + \left( \frac{y}{2} \right)^2 = 1 \]

Step 4: Simplify the equation

Expand the squares and simplify:\[ \frac{x^2}{16} + \frac{y^2}{4} = 1 \]Multiply through by 16 to eliminate the fractions:\[ x^2 + 4y^2 = 16 \]

Step 5: Identify the type of conic section

The final equation \( x^2 + 4y^2 = 16 \) is the rectangular form of the parametric equations. This equation represents an ellipse, with the major axis along the x-direction.

Key Concepts

Trigonometric Identities

In trigonometry, identities are equations involving trigonometric functions that hold true for all values of the involved variables. They are crucial in simplifying and transforming trigonometric expressions.
In the context of parametric equations, two identities come into play most often: the Pythagorean identity and angle sum and difference identities.

• These identities help express trigonometric variables like \( \cos t \) and \( \sin t \) in terms of other variables.
• They allow the conversion of parametric equations into algebraic or rectangular forms, which can be represented as a single equation.

Understanding trigonometric identities is essential to move seamlessly from parametric to rectangular equations.

Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry. It states that the sum of the squares of sine and cosine of an angle equals one:
\[ \cos^2 t + \sin^2 t = 1 \]
This identity emerges from the Pythagorean theorem applied to a unit circle, where any point \((x, y)\) on the circle can be described by \( \cos t \) and \( \sin t \).

• In problems involving parametric equations, the identity helps in eliminating parameters to form a relationship between \(x\) and \(y\).
• By expressing \(\cos t\) and \(\sin t\) in terms of \(x\) and \(y\), you can directly substitute into the Pythagorean identity to simplify to a rectangular equation.

This technique helps transition smoothly from parametric form to a more familiar Cartesian or rectangular form.

Rectangular Equation

A rectangular equation is a standard algebraic representation of a curve in a two-dimensional plane using x and y coordinates. In our problem, we derive it from parametric equations.
To convert parametric equations into a rectangular format:

• First, isolate trigonometric functions like \( \cos t \) and \( \sin t \).
• Next, substitute these expressions into a relevant identity, often the Pythagorean identity.
• Finally, manipulate the resulting equation to express the relationship between \(x\) and \(y\) free of parameters.

This conversion simplifies graphing and analyzing the curve, as rectangular forms are more straightforward to handle without referring to an additional parameter, such as \(t\).

Ellipse

Ellipses are smooth, oval-shaped curves in the coordinate plane that have many applications in physics and astronomy. In the context of a rectangular equation, an ellipse is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
where \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes, respectively.

• In our example, the equation \( x^2 + 4y^2 = 16 \) is in a form that represents an ellipse.
• After simplifying, it shows the ellipse's major axis is along the x-direction as \(\frac{1}{16}x^2\) shows a longer spread compared to \(\frac{1}{4}y^2\).
• The constants can be equated to understand the lengths of axes straight from the equation: 16 as semi-major, scaled appropriately, and 4 as semi-minor.

Understanding how an ellipse takes shape in rectangular form helps in visualizing and graphing these curves in practical scenarios.