Question

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

Short answer

The equation without the parameter is \( x = 1 - 2y^2 \).

Step-by-step solution

Express x in terms of cosine

We have the parametric equation for x as \( x = \cos (2t) \). This can be expanded using the double angle identity: \( \cos (2t) = \cos^2(t) - \sin^2(t) \).

Solve y for sine

We know from the parametric equation that \( y = \sin(t) \). Thus, \( \sin(t) = y \). This gives us the substitution to use in the next step.

Substitute in terms of sine

Using the identity for cosine from Step 1 and substituting \( \sin(t) = y \) into \( x = \cos^2(t) - \sin^2(t) \), we get \( x = \cos^2(t) - y^2 \).

Express x using a Pythagorean Identity

Using the Pythagorean identity \( \cos^2(t) = 1 - \sin^2(t) \), substitute \( \cos^2(t) = 1 - y^2 \) into the expression from Step 3. So, \( x = (1 - y^2) - y^2 = 1 - 2y^2 \).

Final Equation without Parameter

The eliminated parameter equation becomes \( x = 1 - 2y^2 \), which represents the relation between x and y without involving the parameter t.

Key concepts

Eliminating Parameters

Parametric equations often involve an extra variable known as a parameter, which adds a layer of complexity. The task of eliminating the parameter simplifies the equation by expressing the variables directly in terms of each other, typically in Cartesian form. This procedure generally requires using known identities or substitutions to remove the parameter.

Firstly, identify the given equations. In our exercise, we start with the parametric forms \(x = \cos(2t)\) and \(y = \sin(t)\). The goal is to express \(x\) purely in terms of \(y\), effectively removing \(t\).

One common method is to solve one equation for the parameter, then substitute it into the other equation. With \(y = \sin(t)\), we can express \(\sin(t)\) as \(y\). Using this substitution step-by-step helps break down the complexity, leading to \(x = 1 - 2y^2\), which is free from the parameter \(t\) and demonstrates the relationship between \(x\) and \(y\).

Trigonometric Identities

Trigonometric identities are vital tools in working with parametric equations. They allow us to transform expressions and facilitate substitutions.

In the exercise, the double angle identity for cosine is used: \(\cos(2t) = \cos^2(t) - \sin^2(t)\). This identity helps transform the expression involving the parameter, making it easier to eliminate.

Let's go step by step:

• Double Angle Identity: Converts \(\cos(2t)\) into a combination of \(\cos(t)\) and \(\sin(t)\), which is essential for parameter elimination.
• Substitution: Replacing \(\sin(t)\) with \(y\) directly connects the trigonometric function with a known variable, simplifying further computation.

Understanding these identities and using them effectively reduces parametric equations, simplifying complex trigonometry tasks.

Pythagorean Identity

The Pythagorean identity is one of the basic trigonometric identities and plays a crucial role in simplifying equations.

In this exercise, it states that \(\cos^2(t) + \sin^2(t) = 1\). This relationship allows us to express either sine or cosine in terms of the other, which is particularly useful when solving Parametric Equations.

Here's how it was applied in the solution:

• Since \(\sin(t) = y\), it follows that \(\sin^2(t) = y^2\).
• Using the identity \(\cos^2(t) = 1 - \sin^2(t)\) simplifies to \(\cos^2(t) = 1 - y^2\).

Substitute this in the expression for \(x\) to obtain \(x = 1 - 2y^2\). This showcases how the Pythagorean identity helps transition from parametric to a non-parametric equation, effectively eliminating the parameter.