Problem 29 - Calculus

Question

Convert \(x+y=1\) into an \(r \theta\) equation and solve tor \(r .\) Then substitute this \(r\) into \(x=r \cos \theta\) and \(y=r \sin \theta\) to find parametric equations for the line.

Step-by-Step Solution

Answer

Parametric equations are \( x = \frac{\cos\theta}{\cos\theta + \sin\theta} \) and \( y = \frac{\sin\theta}{\cos\theta + \sin\theta} \).

Step 1: Set up Polar Coordinate Conversion

To start, we know that in polar coordinates, the relationships between the Cartesian coordinates and polar coordinates are given by:\[ x = r \cos(\theta) \] \[ y = r \sin(\theta) \]Substitute these into the given equation \(x+y=1\).

Step 2: Substitute and Simplify

Substitute \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\) into the equation:\[ r \cos(\theta) + r \sin(\theta) = 1 \]Factor out the common \(r\):\[ r(\cos(\theta) + \sin(\theta)) = 1 \]

Step 3: Solve for r

To find \(r\), divide both sides of the equation by \((\cos(\theta) + \sin(\theta))\):\[ r = \frac{1}{\cos(\theta) + \sin(\theta)} \]

Step 4: Substitute r into x and y Formulas

Now, use the expression for \(r\) to find the parametric equations. Substitute \(r = \frac{1}{\cos(\theta) + \sin(\theta)}\) into \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\):\[ x = \left( \frac{1}{\cos(\theta) + \sin(\theta)} \right) \cos(\theta) \]\[ y = \left( \frac{1}{\cos(\theta) + \sin(\theta)} \right) \sin(\theta) \]

Step 5: Simplify Parametric Equations

Simplify the expressions for \(x\) and \(y\):\[ x = \frac{\cos(\theta)}{\cos(\theta) + \sin(\theta)} \]\[ y = \frac{\sin(\theta)}{\cos(\theta) + \sin(\theta)} \]

Key Concepts

Parametric Equations

In mathematics, when we talk about parametric equations, we refer to a way of expressing a set of related quantities as explicit functions of an independent variable. Instead of representing geometric figures (like lines or circles) with an equation involving only the standard coordinates (like x and y), a new variable known as a parameter is introduced.

For example, a line in the Cartesian plane can be expressed by using a single parameter \(t\) such as:

• \(x = x_0 + at\)
• \(y = y_0 + bt\)

where \(x_0, y_0\) are initial points and \(a, b\) are direction vectors.

In the context of the exercise, after converting from polar to parametric form, the line \(x + y = 1\) was expressed using equations with parameter \(\theta\). These equations allow us to represent the line in a way that's independent of specific position coordinates, using polar components. This helps in areas like computer graphics, where parametrically driven animations are common.

Polar Form Conversion

Converting between Cartesian and polar coordinates is an essential skill in mathematics, especially when dealing with circular shapes or radar plots. The process involves using the trigonometric relationships to represent a Cartesian point \((x,y)\) as a polar point \((r, \theta)\).

The fundamental formulae involved are:

• \( x = r \cos(\theta) \)
• \( y = r \sin(\theta) \)

When tasked with converting an equation like \(x+y=1\) to a polar form, these relationships allow us to express both \(x\) and \(y\) in terms of \(r\) and \(\theta\). After substitution, solving for \(r\) involves simple algebraic manipulation to isolate \(r\) on one side of the equation.

Such conversions are useful in various applications, ranging from physics (waves and oscillations) to engineering (signal processing). They provide a different perspective that is more convenient in certain mathematical problems, especially when symmetry around the origin is involved.

Cartesian Coordinates

Cartesian coordinates serve as the foundation for two-dimensional space representations. They use a pair of numerical coordinates, which can show spatial location in relation to two perpendicular lines, traditionally called the x-axis and the y-axis.

This system is named after René Descartes, who laid the groundwork for modern day analytic geometry. In basic terms, every point on the plane is defined by an x (horizontal) and y (vertical) value. Lines like \(x+y=1\) are expressed directly using such coordinates by assessing intersections and slopes.

The Cartesian coordinate system is integral to a variety of fields:

• Data visualization (graphs and charts)
• Engineering and architecture (blueprints)
• Physics (motion trajectories)

While relatively simple, the ability to convert these coordinates to other forms like polar coordinates can provide more insights into problems where radial symmetry or rotational transformations are factors.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the involved variables, thus forming a crucial part of trigonometry involving angles and side ratios in triangles. These identities help in simplifying complex expressions, solving equations, and converting between different forms of functions.

Key trigonometric identities include:

• Pythagorean identity: \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
• Angle sum identity: \( \cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b) \)

In the context of polar expression conversions like \(r = \frac{1}{\cos(\theta) + \sin(\theta)}\), these identities become useful by letting us re-write expressions for computation purposes and ensuring that the expressions maintain equivalence under transformation.

Being comfortable with these identities is essential for solving higher-level mathematics problems in calculus, physics, and engineering, where cyclical phenomena and wave functions are often modeled.