Problem 18 - Precalculus

Question

Convert to polar form. $$x^{2}+y^{2}=25$$

Step-by-Step Solution

Answer

The polar form is \(r = 5\).

Step 1: Identify the Equation

The given equation is in Cartesian form: \(x^2 + y^2 = 25\). This represents a circle with radius \(5\) centered at the origin in the Cartesian plane, as the equation is of the form \(x^2 + y^2 = r^2\).

Step 2: Recall Polar Coordinates Formula

In polar coordinates, any point \((x, y)\) can be expressed using the formulas: \(x = r\cos\theta\) and \(y = r\sin\theta\), where \(r\) is the radius and \(\theta\) is the angle.

Step 3: Substitute Cartesian with Polar Components

Using the formulas from Step 2, replace \(x\) and \(y\) in the equation \(x^2 + y^2 = 25\) with their polar counterparts: \((r\cos\theta)^2 + (r\sin\theta)^2 = 25\).

Step 4: Simplify the Equation

Simplify the expression by using the Pythagorean identity: \((r\cos\theta)^2 + (r\sin\theta)^2 = r^2(\cos^2\theta + \sin^2\theta)\). Since \(\cos^2\theta + \sin^2\theta = 1\), the equation becomes \(r^2 = 25\).

Step 5: Solve for Polar Form

Taking the square root of both sides gives \(r = 5\). Thus, in polar form, the equation is \(r = 5\), indicating a circle of radius 5 centered at the origin with variable angle \(\theta\).

Key Concepts

Cartesian Coordinates

In the world of mathematics, Cartesian coordinates provide a way to describe each point on a plane using two numbers. Named after the French mathematician René Descartes, this system uses a grid defined by two perpendicular axes. These axes are:

• The horizontal axis, known as the x-axis.
• The vertical axis, known as the y-axis.

A point's position in Cartesian coordinates is defined by an ordered pair \( (x, y) \).
Here, "x" represents the horizontal distance from the origin, while "y" represents the vertical distance. If you think of graph paper, Cartesian coordinates tell you where to place a point along the grid lines.
For example, in the equation \(x^2 + y^2 = 25\), the coordinates need to satisfy this relationship to lie on the circle's boundary centered at the origin.

Circle Equation

The circle equation in its standard form is \(x^2 + y^2 = r^2\), where \( r \) is the circle's radius. In the context of Cartesian coordinates:

• The circle is centered at the origin (0,0).
• Each point (x, y) on the circle is exactly r units away from the center.

This equation signifies that every point on the circle forms a right triangle with the x and y coordinates.
Using the equation \(x^2 + y^2 = 25\), we recognize it as a circle with a radius of 5. The relationship shows that the sum of the squares of the distances along the axes equals the square of the radius, defining the perfect round path around the center.

Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry that connects the squares of sine and cosine functions. \(\cos^2\theta + \sin^2\theta = 1\).
This identity is derived from the Pythagorean Theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
When the equation \(x^2 + y^2 = r^2\) is rewritten in polar coordinates as \( (r\cos\theta)^2 + (r\sin\theta)^2 = r^2\), the Pythagorean identity simplifies the expression.

• The term \( \cos^2\theta + \sin^2\theta \) simplifies to 1.
• Consequently, \( r^2 = 25 \) simplifies the equation to focus on the radius alone in the polar form.

This identity proves that trigonometric circles and Cartesian circles match when transforming coordinates.

Radius

The radius is a key concept when dealing with circles. It represents the distance from the center to any point on the circle's edge.
In the context of the equation \(x^2 + y^2 = 25\), the radius is calculated by taking the square root of 25, resulting in a radius of 5.

• All points on the circle maintain this fixed distance of 5 from the center.
• This constant radius forms a boundary, creating a perfect circle.

The idea of radius is not just limited to circles on a plane, but crucial to understanding the geometric properties of shapes, which can help in calculations involving perimeter and area.
When converting to polar form, acknowledging the radius assists in simplifying the circle's equation to \( r = 5 \), effectively capturing the essence of a circle where only the angle changes.