Problem 77 - College Trigonometry

Question

Convert the equation from polar coordinates into rectangular coordinates. $$ r=7 $$

Step-by-Step Solution

Answer

In rectangular coordinates, the equation is \( x^2 + y^2 = 49 \).

Step 1: Understand the Polar Coordinate Equation

The given equation is in polar coordinates: \( r = 7 \). This represents a circle centered at the origin with a radius of 7.

Step 2: Recall the Polar to Rectangular Coordinate Formulas

To convert from polar to rectangular coordinates, use the following formulas: \( x = r \cos \theta \) and \( y = r \sin \theta \).

Step 3: Substitute the Polar Equation into the Rectangular Formulas

Since \( r = 7 \), substitute 7 into the formulas, giving \( x = 7 \cos \theta \) and \( y = 7 \sin \theta \).

Step 4: Use the Pythagorean Identity for Conversion

We are given \( r = 7 \). The rectangular equivalent can be found using the identity \( x^2 + y^2 = r^2 \). Substitute \( r = 7 \), such that \( x^2 + y^2 = 7^2 \).

Step 5: Simplify the Rectangular Equation

Simplify the equation \( x^2 + y^2 = 49 \). This is the rectangular form of the polar equation \( r = 7 \).

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a way to pinpoint locations in a two-dimensional plane using ordered pairs \((x, y)\). Each pair provides an exact location on the plane, where \(x\) represents the horizontal distance from the origin (point \((0, 0)\)), and \(y\) denotes the vertical distance from the origin. This system is named after the mathematician René Descartes and is commonly used in algebra and geometry.In this system, any point on a plane can be described clearly and easily by adjusting the \(x\) and \(y\) values. By plotting various \((x, y)\) points, you can graph lines, shapes, and curves. Using rectangular coordinates allows for straightforward communication and understanding of geometric concepts as well as computer graphics.When converting from other systems like polar coordinates to rectangular coordinates, the main goal is transforming the circular relationship into one based on straight lines and angles.

Polar Coordinates

Polar coordinates offer a different way of identifying points in a plane, using a distance and an angle. Here, a point \((r, \theta)\) describes a location in the following manner:

• \(r\) is the radial coordinate: It represents the distance from the origin (the pole) to the point.
• \(\theta\) is the angular coordinate: It specifies the direction of \(r\) from the positive x-axis, typically measured in radians or degrees.

Polar coordinates are particularly useful for dealing with circles, spirals, and wave forms, providing a way to describe occurrences that would be more complex in rectangular form. Through the relationship with angles and distances, polar coordinates facilitate solving problems where symmetry around a point is significant, such as tracking movement in a circular trajectory.

Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry that relates the sides of a right triangle to its hypotenuse. Expressed as \(\sin^2 \theta + \cos^2 \theta = 1\), it stems from the Pythagorean theorem, \(a^2 + b^2 = c^2\), applied to the unit circle where the hypotenuse is 1.This identity is essential when converting polar coordinates \((r, \theta)\) to rectangular coordinates. For a point located on a circle with radius \(r\), the identity helps to translate angular information (\(\theta\)) into straightforward \(x\) and \(y\) components:

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

Thus, just like that of the triangle's sides, the identity ensures that the contributions of \(x\) and \(y\) emerge naturally and analytically from \(r\) and \(\theta\). This underlies all conversions between polar and rectangular systems, providing a bridge from circular symmetry to linear descriptions.

Coordinate Conversion

Converting between polar and rectangular coordinates is an essential skill in mathematics to depict different geometric and real-world problems efficiently. Here’s how it happens:To change a polar point \((r, \theta)\) to rectangular coordinates \((x, y)\), use:

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

Inversely, to switch from rectangular \((x, y)\) to polar \((r, \theta)\), apply:

• \(r = \sqrt{x^2 + y^2}\)
• \(\theta = \tan^{-1} \left( \frac{y}{x} \right)\)

These conversions serve as a mathematical toolbox for analyzing and understanding movements along curves or within coordinate grids. They make it easier to approach equations triggered by physical scenarios like wave propagation or circular movement, ensuring that any geometric transformation is both precise and repeatable.