Problem 15 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the graph of the given Cartesian equation, and then find the polar equation for it. $$ x^{2}+y^{2}=4 $$

Step-by-Step Solution

Answer

The Cartesian equation describes a circle; its polar form is \(r = 2\).

Step 1: Understand the Cartesian Equation

The given equation is \(x^2 + y^2 = 4\). This represents a circle in the Cartesian coordinate system centered at the origin (0,0) with a radius of 2 because \(\sqrt{4} = 2\).

Step 2: Sketch the Graph of the Circle

Draw the Cartesian coordinate system. Since the equation \(x^2 + y^2 = 4\) is a circle with center (0,0) and radius 2, plot a circle touching the points (2,0), (-2,0), (0,2), and (0,-2) on the coordinate axes.

Step 3: Convert to Polar Coordinates

In polar coordinates, \(x = r\cos\theta\) and \(y = r\sin\theta\). Substitute these into the equation: \((r\cos\theta)^2 + (r\sin\theta)^2 = 4\). This simplifies to \(r^2(\cos^2\theta + \sin^2\theta) = 4\).

Step 4: Simplify Using Trigonometric Identity

Apply the identity \(\cos^2\theta + \sin^2\theta = 1\), yielding \(r^2 = 4\). Solve for \(r\) by taking the square root: \(r = 2\) or \(r = -2\) (though we typically ignore the negative radius in polar form).

Step 5: Write the Polar Equation

The polar equation is \(r = 2\). This indicates a circle in the polar coordinate system with radius 2, centered at the pole (origin).

Key Concepts

Cartesian Equation

A Cartesian equation describes the relationship between the x and y coordinates in a plane, primarily using the rectangular coordinate system. The given equation, \(x^2 + y^2 = 4\), is a quintessential example of a circular equation in Cartesian form. It represents a circle with its center positioned at the origin (0,0).
To understand why, let's break it down:

• The left-hand side, \(x^2 + y^2\), is the sum of squares of the x and y coordinates.
• The right-hand side is a constant, 4, indicating the square of the radius of the circle.

Hence, \(x^2 + y^2 = 4\) implies a circle with a radius of 2, because the square root of 4 is 2.
This formula allows us to represent geometric shapes algebraically, aiding in graphing precise shapes through mathematical equations.

Circle Graph

A circle graph, also known as a circular or polar graph, shows a circle's shape in the Cartesian coordinate system.
When graphing \(x^2 + y^2 = 4\), you follow these steps:

• Firstly, draw the coordinate axes, which usually consist of two perpendicular lines labeled x and y.
• Since the equation tells us that the center is at (0,0) and the radius is 2, plot points exactly 2 units away from the origin in all directions.
• These points lie at (2,0), (-2,0), (0,2), and (0,-2) on the axes.
• Finally, connect these points smoothly to form a circle.

The resulting graph is a perfect circle centered on the origin, perfectly illustrating how algebraic equations shape visual representations of geometric figures.

Trigonometric Identity

Trigonometric identities are equations that hold true for all values of the involved variables, usually involving trigonometric functions. The identity \( \cos^2\theta + \sin^2\theta = 1 \) is one of the most fundamental, detailing that the sum of the square of cosine and the square of sine of any angle is always one.
In the context of converting to polar coordinates:

• The given identity simplifies the Cartesian circle equation\((r\cos\theta)^2 + (r\sin\theta)^2 = 4\).
• By applying \( \cos^2\theta + \sin^2\theta = 1 \), we simplify it to \(r^2 = 4\).

This simplification is crucial in transitioning between Cartesian and polar systems, enabling us to easily understand the shape's geometry in polar coordinates.

Converting Coordinates

Converting between Cartesian and polar coordinates is often necessary to analyze and visualize data in different formats.
For the equation \(x^2 + y^2 = 4\), we can convert it into polar form by using the relationships:

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

Substitute these into the equation, and the conversion process begins:

• The equation \((r\cos\theta)^2 + (r\sin\theta)^2 = 4\) emerges.
• It simplifies via the trigonometric identity to \(r^2 = 4\).
• Taking the square root gives us \(r = 2\), representing the radius in polar form.

This converted equation, \(r = 2\), shows a circle of radius 2 in the polar coordinate system, centered at the pole (origin). Conversions like these help bridge different mathematical representations, enriching our understanding of geometric figures.