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Chapter 6: Problem 38

Eliminate the parameter \(t\) in each of the following: $$x=\cos t, y=-\cos t$$

Short Answer

Expert verified
The equation is \( x = -y \).

Step by step solution

01

Understand the Parametric Equations

The given equations are:\[ x = \cos t \] and \[ y = -\cos t \]. These are parametric equations where both x and y are expressed in terms of a parameter \( t \). Our goal is to eliminate \( t \) and find a single equation involving only \( x \) and \( y \).
02

Express the Parameter in Terms of One Variable

Since both \( x \) and \( y \) involve \( \cos t \), we can start by expressing \( \cos t \) from one of the equations. From \( x = \cos t \), we get:\[ \cos t = x \]Similarly, from \( y = -\cos t \), we can write:\[ \cos t = -y \].
03

Equate the Expressions for \( \cos t \)

Since both expressions represent \( \cos t \), we equate them:\[ x = -y \].This equation now relates \( x \) and \( y \) directly, eliminating the parameter \( t \).
04

Simplify the Equation

The equation \( x = -y \) is already in its simplest form, showing the relationship between \( x \) and \( y \) without involving \( t \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Trigonometric Functions in Parametric Equations
Trigonometric functions like \( \cos t \) and \( \sin t \) are often used in parametric equations to describe circles and oscillatory movements. In the given equations, both \( x \) and \( y \) are expressed in terms of \( \cos t \). This plays a significant role in how the shape or path can be visualized in a coordinate system. Here, \( x = \cos t \) and \( y = -\cos t \) indicate how both quantities are related to the cosine of the same parameter.
  • The function \( \cos t \) varies between -1 and 1 as \( t \) changes, providing all possible values for \( x \) and \( y \) within this range.
  • Since cosine is symmetrical, the expressions hint at how \( x \) and \( y \) could be mirror images across the origin depending on the values of \( t \).
Recognizing that \( x \) and \( y \) both stem from the same trigonometric function is essential for understanding the geometric interpretation of these parametric equations.
Expressing Parameters in Terms of a Known Quantity
To find a direct relationship between \( x \) and \( y \) without involving the parameter \( t \), we need to express the parameter using one of the equations. In this exercise, expressing \( \cos t \) is straightforward since the two parametric equations are linear in form. Here are the steps:
  • First, take \( x = \cos t \) and solve for \( \cos t \), which is already expressed as \( \cos t = x \).
  • Next, do the same for \( y = -\cos t \), giving us \( \cos t = -y \).
By expressing \( \cos t \) first from \( x \) and then from \( y \), you can create a link between these two variables, setting the stage for elimination of \( t \). The essence of this process is isolating the trigonometric function that ties both equations together.
Eliminating Parameters to Find Relationships
The goal of eliminating a parameter is to derive an equation that only involves the variables \(x\) and \(y\). This results in a better understanding of the relationship between the two variables. With the expressions for \( \cos t \) from \( x \) and \( y \) being equal, we achieve this goal by equating these expressions:
  • Starting from \( \cos t = x \) and \( \cos t = -y \), set \( x = -y \).
  • This simple equation now shows how \( x \) is directly related to \( y \) without any mention of \( t \).
  • The relationship \( x = -y \) simplifies the problem and explains how the two variables counterbalance each other, completing the instructions without needing additional parameter information.
Eliminating parameters is a powerful technique that simplifies parametric equations into a format that shows direct linkage between variables, enhancing clarity and interpretation of results.

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Most popular questions from this chapter

Use your graphing calculator to find all radian solutions in the interval \(0 \leq x<2 \pi\) for each of the following equations. Round your answers to four decimal places. $$ \cos ^{2} x-3 \cos x+1=0 $$

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$$ \text { Find all degree solutions. } $$ $$ 2 \sin ^{2} 3 \theta+3 \sin 3 \theta+1=0 $$

$$ \text { Find all degree solutions. } $$ $$ \cot ^{2} 3 \theta=1 $$
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