Problem 42 - Contemporary Precalculus

Question

Find a rectangular equation that is equivalent to the given polar equation. $$r=5$$

Step-by-Step Solution

Answer

Question: Convert the polar equation r = 5 into a rectangular equation. Answer: The rectangular equation equivalent to the given polar equation is x^2 + y^2 = 25.

Step 1: Rewrite the polar equation as a rectangular equation

Given the polar equation $$r = 5$$, we'll write this equation in terms of x and y using the polar to rectangular coordinate conversion formulas $$x = r\cos (\theta)$$ and $$y = r\sin (\theta)$$. Firstly, let's square both sides of the polar equation: $$(r)^2 = (5)^2$$ Now, let's square the conversion formulas for x and y: $$x^2 = (r\cos (\theta))^2$$ $$y^2 = (r\sin (\theta))^2$$ And then add the squared conversion formulas: $$x^2 + y^2 = (r\cos (\theta))^2 + (r\sin (\theta))^2$$ Then use the Pythagorean identity to simplify: $$x^2 + y^2 = r^2(\cos^2 (\theta) + \sin^2 (\theta))$$ Since $$\cos^2 (\theta) + \sin^2 (\theta) = 1$$, then: $$x^2 + y^2 = r^2$$

Step 2: Substitute the polar equation into the rectangular equation

Finally, substitute the squared polar equation, $$r^2 = 5^2 = 25$$, into the rectangular equation: $$x^2 + y^2 = 25$$ This is the rectangular equation that is equivalent to the given polar equation.

Key Concepts

Rectangular Equation

In the world of mathematics, particularly in coordinate systems, the rectangular equation refers to an equation expressed in terms of rectangular or Cartesian coordinates, which are denoted as 'x' and 'y'. These are the types of equations most students encounter first, as they resemble the standard graphing format taught in algebra. An example of a simple rectangular equation is the equation of a circle, which can be written as:
\[x^2 + y^2 = r^2\]
where 'r' stands for the radius of the circle. To find a rectangular equation equivalent to a polar equation, we set out to eliminate the polar variables (r and \(\theta\)) and describe the relationship purely in terms of 'x' and 'y'. This is particularly useful for plotting the curve on a standard Cartesian plane or for integrating functions over areas that are easier to describe in Cartesian coordinates.

Polar Equation

Conversely, a polar equation is one that is described in terms of polar coordinates 'r' and \(\theta\). Here, 'r' represents the radial distance from the origin to a point on the curve, and \(\theta\) represents the angle formed between the positive x-axis and the line segment from the origin to the point. In the case of the given exercise, the polar equation is remarkably simple:
\[r = 5\]
This equation describes a circle with a radius of 5 units centered at the origin in a polar coordinate system. Unlike rectangular equations, polar equations often provide a more natural and straightforward description of curves with circular or spiral shapes.

Pythagorean Identity

Central to converting from polar to rectangular coordinates is the Pythagorean identity, which is a cornerstone of trigonometry. It states that for any angle \(\theta\):
\[\text{cos}^2(\theta) + \text{sin}^2(\theta) = 1\]
It originates from the Pythagorean Theorem that applies to right-angled triangles. This identity helps in simplifying polar equations that involve sine and cosine functions before converting them into rectangular form. By applying the Pythagorean identity, you're able to express the sum of the squares of 'x' and 'y' as the square of the radius 'r', thus enabling the transition from the circular polar concept to the more linear Cartesian framework.

Coordinate Conversion

The process of transitioning from one coordinate system to another is known as coordinate conversion. When converting from polar to rectangular coordinates, we utilize two important equations:
\[x = r\cdot\text{cos}(\theta)\]\[y = r\cdot\text{sin}(\theta)\]
These formulas are derived from the definitions of the sine and cosine functions. By squaring both 'x' and 'y' and adding them, we can make use of the Pythagorean identity to eliminate the \(\theta\) variable and obtain a rectangular equation in terms of 'x' and 'y' alone. This process is essential for analyzing and graphing equations in the most suitable coordinate system, thereby enhancing our understanding and simplifying computations.