Problem 30 - Contemporary Precalculus

Question

The given curve is part of the graph of an equation in \(x\) and \(y .\) Find the equation by eliminating the parameter. $$x=4 \sin 2 t, \quad y=2 \cos 2 t, \quad 0 \leq t \leq 2 \pi$$

Step-by-Step Solution

Answer

Answer: The relationship between \(x\) and \(y\) for the given parametric equations is \(x^2 + 2y^2 = 4\).

Step 1: Solve for \(\sin 2t\) and \(\cos 2t\) from the given equations

The given parametric equations can be written as: $$\sin 2t = \frac{x}{4} \quad \text{and} \quad \cos 2t = \frac{y}{2}$$ We need to find the relationship between \(x\) and \(y\) by using the trigonometric identity that connects \(\sin\) and \(\cos\).

Step 2: Use the Pythagorean identity to eliminate \(t\)

The Pythagorean identity for sine and cosine states that for any angle \(\theta\): $$\sin^2 \theta + \cos^2 \theta = 1$$ So, we can apply this for our angle \(2t\): $$(\sin 2t)^2 + (\cos 2t)^2 = 1$$ Now, substitute the expressions we found in Step 1: $$\left(\frac{x}{4}\right)^2 + \left(\frac{y}{2}\right)^2 = 1$$

Step 3: Simplify the equation

To eliminate the fractions, we can multiply both sides of the equation by the least common multiple of the denominators, which is 4: $$4 \cdot \left(\frac{x}{4}\right)^2 + 4 \cdot \left(\frac{y}{2}\right)^2 = 4 \cdot 1$$ Simplifying the equation, we get: $$x^2 + 2y^2 = 4$$

Step 4: Write the final equation

Rearranging the equation found in Step 3, we arrive at the final equation relating \(x\) and \(y\): $$x^2 + 2y^2 = 4$$

Key Concepts

Parametric Equations

Parametric equations are a way to define a set of equations that express the coordinates of points on a curve as functions of a parameter, usually denoted as "t". In the given problem, we have two parametric equations: \( x = 4 \sin 2t \) and \( y = 2 \cos 2t \).

Parametric equations have the advantage of being able to represent a wider variety of curves compared to regular Cartesian equations. Instead of directly associating \( x \) and \( y \) with one another, they allow for independent representation through an intermediary variable or parameter.

Being comfortable with parametric equations is particularly useful when dealing with motion problems and in situations where both \( x \) and \( y \) depend simultaneously and independently on a third variable. Converting these into a single equation involving only \( x \) and \( y \) is known as eliminating the parameter; this helps to see the relationship between the two variables directly.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all possible values of the variables involved. A key component in solving problems involving circular motion or oscillations is these identities, as they can be employed to simplify complex trigonometric expressions.

In this exercise, we utilized one of the basic trigonometric identities: the identity \( \sin 2t = \frac{x}{4} \) and \( \cos 2t = \frac{y}{2} \) were used to eliminate the parameter \( t \).

• The identities used are fundamental tools to manipulate equations.
• They can transform trigonometric expressions into relations involving algebraic expressions.

Additionally, these identities are critical when converting between parametric and non-parametric equations, helping us understand and manipulate the relationships between different geometric entities.

Pythagorean Identity

The Pythagorean identity is one of the most crucial and commonly used identities in trigonometry. It is given by \( \sin^2 \theta + \cos^2 \theta = 1 \) for any angle \( \theta \). This identity expresses a fundamental relationship between the sine and cosine of an angle, analogous to the Pythagorean theorem.

In this exercise, the Pythagorean identity was crucial for eliminating the parameter and finding the relationship between \( x \) and \( y \). By substituting \( \sin 2t = \frac{x}{4} \) and \( \cos 2t = \frac{y}{2} \) into the Pythagorean identity:
\[\left(\frac{x}{4}\right)^2 + \left(\frac{y}{2}\right)^2 = 1\]

• We can directly relate the parametric variables to Cartesian variables.
• This form helps to recognize shapes like ellipses or circles from the equations.

Finally, this manipulation results in a simplified equation \(x^2 + 2y^2 = 4\), showing that trigonometric identities such as the Pythagorean identity provide a powerful bridge between different mathematical representations.