Problem 49 - Precalculus Mathematics for Calculus

Question

Convert the polar equation to rectangular coordinates. $$r=7$$

Step-by-Step Solution

Answer

The rectangular form is \( x^2 + y^2 = 49 \).

Step 1: Understand Polar and Rectangular Coordinates

The given equation is in polar coordinates: \( r = 7 \). In polar coordinates, \( r \) represents the radial distance from the origin, and an angle (\( \theta \)) usually characterizes the direction. To convert this into rectangular coordinates \((x, y)\), we need to use the relationships: \( x = r \cos \theta \) and \( y = r \sin \theta \).

Step 2: Express \( r \) in Terms of \( x \) and \( y \)

The relationship between polar and rectangular coordinates also includes a fundamental equation where \( r = \sqrt{x^2 + y^2} \). Given \( r = 7 \), we can substitute \( 7 \) for \( r \) to get \( 7 = \sqrt{x^2 + y^2} \).

Step 3: Remove Square Root by Squaring Both Sides

To isolate \( (x, y) \), we square both sides of the equation in Step 2: \( (7)^2 = (\sqrt{x^2 + y^2})^2 \). This simplifies to \( 49 = x^2 + y^2 \).

Step 4: Conclusion for Conversion

The polar equation \( r = 7 \) in rectangular coordinates is the equation of a circle with a radius of 7, centered at the origin. Thus, the rectangular form is \( x^2 + y^2 = 49 \).

Key Concepts

Rectangular Coordinates

Rectangular coordinates form an essential part of our understanding of geometry and algebra. They are often used to define the position of a point in the two-dimensional plane. The most common system we work with is the Cartesian coordinate system, characterized by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Each point in this plane is represented as an ordered pair \(x, y\), where \(x\) describes the horizontal positioning and \(y\) describes the vertical.The conversion between different coordinate systems is crucial for solving various mathematical problems. For instance, when working with equations or shapes defined in polar coordinates, translating them into rectangular form can simplify calculations or make them suitable for certain types of graph analysis. Understanding the relationships and transformations between these systems expands our toolkit for approaching mathematical challenges.

Polar Coordinates

Polar coordinates offer an alternative way to describe a point on the plane, based more on distances and angles than right-angle measures. They are particularly useful in contexts where rotational symmetry or circular paths are involved, such as physics or engineering problems.In polar coordinates, any point on the plane is described by two values: \(r\) and \(\theta\).

• \(r\) is the radial distance from the origin (usually denoted as the center of the system).
• \(\theta\) is the angular position, measured in radians or degrees, relative to the positive x-axis.

To convert from polar to rectangular form, we use the relationships \(x = r \cos \theta\) and \(y = r \sin \theta\). These formulas enable us to express any polar coordinate \(r, \theta\) as an equivalent rectangular coordinate \(x, y\), facilitating easier plotting or interpreting on Cartesian grids.

Equation of a Circle

Understanding the equation of a circle is vital for anyone dealing with geometry or algebraic representations of round shapes. In mathematics, a circle is defined by its center point and radius.The standard equation for a circle in rectangular coordinates is:\[x^2 + y^2 = r^2\]where \(r\) is the radius of the circle, and the circle is centered at the origin \(0,0\).In our exercise, we started with a polar equation \(r = 7\). By transforming this into rectangular form, we derived \(x^2 + y^2 = 49\), which corresponds to a circle around the origin with a radius of 7.A good grasp of how different forms of equations represent the same shape helps in transitioning between coordinate systems or solving geometric problems, giving a deeper insight into their behavior and properties.