Problem 94 - Precalculus with Limits

Question

In Exercises 85-108, convert the polar equation to rectangular form. \(r=10\)

Step-by-Step Solution

Answer

The rectangular form of the polar equation \(r=10\) is \(x^2 + y^2 = 100\).

Step 1: Understanding Polar Coordinates

Polar coordinates are represented as \(r,\theta\), where r represents the distance from the origin and \(\theta\) represents the angle measured in a counterclockwise direction from the positive x-axis. Any point in polar coordinates can be mapped into an equivalent point in Cartesian coordinates using the relationships \(x = rcos(\theta)\) and \(y = rsin(\theta)\).

Step 2: Convert Polar to Rectangular

For the equation given, \(r=10\), we do not have an angle \(\theta\). However, we do know that regardless of what \(\theta\) is, the distance from the origin is always 10. This means that we are dealing with a circle centered at the origin with radius 10. The Cartesian equation for a circle centered at the origin is \(x^2 + y^2 = r^2\).

Step 3: Substitution

Now we need to substitute \(r=10\) into the rectangular form equation from Step 2. When we substitute \(r=10\) into \(x^2 + y^2 = r^2\), we get \(x^2 + y^2 = 10^2\), which simplifies to \(x^2 + y^2 = 100\). This is the rectangular form of the polar equation \(r=10\).

Key Concepts

Polar Coordinates

Polar coordinates offer an alternative way to describe the location of a point in a plane. Imagine standing at the origin of a grid. Polar coordinates use two values to tell you how far and in what direction to move.

• The first value, \(r\), is the radial distance from the origin. Think of it as how many steps you take forward.
• The second value, \(\theta\), is the angle measured from the positive x-axis in a counterclockwise direction. It tells you which way to turn before you start walking.

By using the equations \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\), you can translate polar coordinates to rectangular coordinates, which most people find more familiar. This transformation is useful because many problems in mathematics and physics can be more easily solved in the system that best fits their nature.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, use pairs \((x, y)\) to describe points on a plane. This method involves creating a grid with perpendicular axes. From the origin, you can:

• Move horizontally to find the \(x\)-coordinate.
• Move vertically to reach the \(y\)-coordinate.

This system is intuitive because it resembles a city map, with streets forming perpendicular intersections. It is particularly useful in problems requiring calculations of distance or plotting graphs. For example, once you know \(x\) and \(y\), you can easily calculate the distance from the origin using the simple formula \(\sqrt{x^2 + y^2}\). Additionally, you can convert from polar coordinates to rectangular coordinates to harness this practicality.

Cartesian Equation of Circle

The Cartesian equation for a circle centered at the origin in rectangular coordinates is an elegant expression. A circle can be defined by its center and its radius, and for a circle at the origin, the equation is \[ x^2 + y^2 = r^2 \]where \(r\) is the radius of the circle.

• The equation represents all the points \((x, y)\) that are exactly \(r\) units away from the origin.
• It makes it easy to understand complex shapes by viewing the circle as an arrangement of points equidistant from a central point.

For instance, the polar equation \(r = 10\) converts simply into the rectangular form \(x^2 + y^2 = 100\) following the idea that a circle with radius 10 units comprises all points 10 units from the center, fulfilling the requirement in rectangular terms as well.