Problem 17 - Algebra and Trigonometry

Question

Sketch a graph of the polar equation, and express the equation in rectangular coordinates. $$ r=2 $$

Step-by-Step Solution

Answer

It's a circle with radius 2, centered at the origin: \( x^2 + y^2 = 4 \).

Step 1: Identifying the Polar Equation

We begin with the given polar equation \( r = 2 \). This equation states that the radius \( r \) from the origin to any point on the graph is always 2. This means that every point on the graph is 2 units away from the origin.

Step 2: Sketching the Polar Graph

Since \( r = 2 \) is a constant, the graph in polar coordinates is a circle centered at the origin with a radius of 2. Sketch a circle with its center at the origin (0,0) and passing through (2,0) and (-2,0). The circle extends equally in all directions from the center (origin) of the polar plane.

Step 3: Converting to Rectangular Coordinates

The polar equation \( r = 2 \) can be converted into rectangular coordinates. In polar coordinates, \( r^2 = x^2 + y^2 \). Since \( r = 2 \), substituting gives \( (2)^2 = x^2 + y^2 \). Thus, the equation in rectangular coordinates is \( x^2 + y^2 = 4 \).

Step 4: Final Rectangular Equation

The expression \( x^2 + y^2 = 4 \) represents a circle in rectangular (Cartesian) coordinates with a center at the origin (0,0) and a radius of 2, similar to the polar graph's interpretation.

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a way to define a point in a plane with a pair of numerical values. These values, usually noted as \( (x, y) \), correspond to the horizontal and vertical distances from a fixed point called the origin. Understanding this system helps in graphing geometric shapes and solving mathematical problems involving space.

• The horizontal axis is labeled as the \( x \)-axis.
• The vertical axis is labeled as the \( y \)-axis.
• The point where these two axes intersect is the origin, noted as \( (0, 0) \).

In the context of the original exercise, converting from polar to rectangular involves expressing a circle in these terms. When we say \( x^2 + y^2 = 4 \), it describes a circle centered at the origin with a radius of 2. This is because both \( x \) and \( y \) are positioned such that any point that satisfies the equation is equally distanced from the center, forming a perfect circle.

Graphing Polar Equations

Graphing polar equations involves plotting points based on their distance from the origin and their angle relative to a fixed line. Unlike rectangular coordinates that use a grid, polar coordinates use a circular system.In polar coordinates:

• \( r \) represents the radius or distance from the origin.
• \( \theta \) represents the angle from a fixed direction, typically the positive \( x \)-axis.

To graph \( r = 2 \), you visualize all points which are exactly 2 units away from the origin. This creates a circle with a radius of 2. The center of the circle is at the origin, and the radius extends outward in all directions equally, meaning that every point is consistently 2 units away, resulting in a perfect circle. This concept is quite distinct from plotting in the Cartesian plane, adding a dynamic perspective with angles and distances.

Converting Equations

Converting equations between polar and rectangular forms is crucial in understanding complex geometry and simplifying calculations. Each form has its own advantages, like clarity in different situations.To convert an equation from polar to rectangular:

• Use the fundamental relationship \( r^2 = x^2 + y^2 \). This is derived from the Pythagorean theorem applied to the polar coordinate system.
• Substitute \( r \) and \( \theta \) values using known trigonometric relationships, such as \( x = r \cos \theta \) and \( y = r \sin \theta \).

In the given exercise, the original polar equation \( r = 2 \) was turned into \( x^2 + y^2 = 4 \). This highlights the conversion process whereby the geometry remains consistent even as the coordinate system changes. Understanding these conversions allows deeper insight into the graphical representation of equations and enhances problem-solving capabilities.