Problem 93 - Precalculus with Limits

Question

In Exercises 85-108, convert the polar equation to rectangular form. \(r=4\)

Step-by-Step Solution

Answer

The rectangular form of the polar equation \(r=4\) is \(x = 4 \cos \theta\), \(y = 4 \sin \theta\), given that \(\sqrt{x^2+y^2} = 4\).

Step 1: Conversion from Polar to Rectangular Coordinates

Given the simple polar equation \(r=4\), no specific angle theta is mentioned. The conversion from polar to rectangular form uses the formulas \(x=r \cos \theta\) and \(y=r \sin \theta\). For this equation, because r = 4, we can substitute r into these formulas to derive the rectangular form.

Step 2: Equation for x

By substituting r into \(x=r \cos \theta\), we get \(x = 4 \cos \theta\). This cannot be converted any further because a specific theta is not given.

Step 3: Equation for y

By substitifying r into \(y=r \sin \theta\), we get \(y = 4 \sin \theta\). This too cannot be converted any further because a specific theta is not given.

Step 4: Rectangular Form

The conversion of \(r=4\) into rectangular form would therefore result in the equations \(x = 4 \cos \theta\) and \(y = 4 \sin \theta\), provided that \(\sqrt{x^2+y^2} = 4\) because \(r= \sqrt{x^2+y^2}\) and r = 4 as per the provided polar equation.

Key Concepts

Polar Coordinates

Polar coordinates are a way to locate a point in a plane using two values: the radius and the angle. Unlike the usual way of graphing points with x and y coordinates, polar coordinates use

• the radius (\(r\)) - the distance from the origin to the point, and
• the angle (\(\theta\)) - measured from the positive x-axis, typically in radians or degrees.

This system helps to describe a point's position in a circular manner, revolving around the origin. For example, the polar coordinate \((r, \theta) = (4, 45^{\circ})\) means a point is 4 units away from the origin at an angle of 45 degrees relative to the positive x-axis. In cases where the angle isn't specified, as in the exercise with \(r=4\), the point is somewhere on the circle of radius 4.

Rectangular Coordinates

Rectangular coordinates, or Cartesian coordinates, describe the position of a point with two values: x and y. These values indicate how far a point is from the origin along the x-axis and y-axis:

• \(x\) - represents the horizontal distance from the origin, and
• \(y\) - represents the vertical distance from the origin.

This system is straightforward for most graphing and algebra purposes as it lays out points in a grid-like manner. Every point in a rectangular coordinate system can be written as \((x,y)\). For example, if a point is 3 units to the right and 2 units up from the origin, its coordinates are \((3,2)\). The relationship of a shape or figure in this system is often easy to visualize with your usual xy-plane.

Conversion Formulas

Conversion between polar and rectangular coordinates involves specific formulas. The goal is to express a point or equation known in one system, like polar, in the other, like rectangular, and vice versa. The conversion formulas are:

• From polar to rectangular:
  • \(x = r \cos \theta\)
  • \(y = r \sin \theta\)
• From rectangular to polar:
  • \(r = \sqrt{x^2 + y^2}\)
  • \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\)

These formulas work by relating the sides of a right triangle formed by the point's distance and angles. For example, if \(r=4\) without a specific \(\theta\), we can only express \(x\) and \(y\) in terms of \(\cos\) and \(\sin\) using the given radius. Hence, the equation becomes \(\sqrt{x^2 + y^2} = 4\), which is the essential link between the polar and rectangular representations.