Problem 74 - Precalculus

Question

Convert the rectangular equation to polar form. Assume \(a > 0\). $$y=-x$$

Step-by-Step Solution

Answer

Therefore, the polar form of the rectangular equation \(y=-x\) is \(θ = 3/4 π or 7/4 π\).

Step 1: Write Down the Relations

Before doing any transformations, it's important to know the relations between rectangular and polar coordinates. They can be defined as: \(x = rcos(θ)\) and \(y = rsin(θ)\). So any 'x' found can be replaced by \(rcos(θ)\) and 'y' would be replaced by \(rsin(θ)\)

Step 2: Substitute the Equations

Making substitution in the given rectangular equation \(y = -x\) using the related polar equations. That will make it \(rsin(θ) = - rcos(θ)\)

Step 3: Simplify Equation

Solving for r we get \(rsin(θ) + rcos(θ) = 0\), factoring out r we get \(r(sincos(θ) + cos(θ)) = 0\). This implies either \(r = 0\) or \(tan(θ) = -1\). In polar coordinates, \(r\) can be any real number and \(θ\) usually lies from \(0\) to \(2π\), so using the \(tan(θ) = -1\) part, we get \(θ = 3/4 π or 7/4 π\).

Key Concepts

Polar Coordinates

Polar coordinates are a two-dimensional coordinate system where each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point, known as the pole, corresponds to the origin in the rectangular coordinate system, and the reference direction, typically the positive x-axis, is called the polar axis.

In polar coordinates, the location of a point is given as \( r, \theta \), where \( r \) represents the radial distance from the pole (origin), and \( \theta \) is the angular component, measured in radians from the polar axis. It's a unique system that's particularly useful for dealing with problems involving curves and angles, such as those encountered in physics and engineering.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are represented by two values, usually \( x \) and \( y \) on a two-dimensional plane. Each value corresponds to a position along horizontal and vertical axes, respectively. In this system, any point in a plane can be described by its horizontal distance from the y-axis and its vertical distance from the x-axis.

This coordinate system is widely used in algebra and calculus as it simplifies many types of problems, especially those involving straight lines and polynomials. Converting from rectangular to polar coordinates requires the application of trigonometric identities, which relate the angles and distances of a point in both systems.

Trigonometric Identities

Trigonometric identities are equations that relate the trigonometric functions (sine, cosine, tangent, etc.) to one another. They are based on the properties of a right triangle but apply to any angle. These identities are crucial for simplifying expressions and solving equations involving trigonometric functions.

Some common trigonometric identities include the Pythagorean identities, such as \( \sin^2(\theta) + \cos^2(\theta) = 1 \), and angle sum and difference identities. When converting rectangular coordinates to polar coordinates, trigonometric identities are used to express \( x \) and \( y \) in terms of \( r \) and \( \theta \) by setting \( x = r\cos(\theta) \) and \( y = r\sin(\theta) \) as a direct application of these identities.