Question

Convert the rectangular equation to polar form and sketch its graph. $$ x^{2}+y^{2}=16 $$

Short answer

The polar form is \( r = 4 \), which is a circle of radius 4 centered at the origin.

Step-by-step solution

Identify the Rectangular Equation

The given rectangular equation is \( x^2 + y^2 = 16 \). This is a standard form of a circle centered at the origin with radius 4.

Understand the Polar Coordinate Relationship

In polar coordinates, any point \((x, y)\) can be represented as \((r \cos \theta, r \sin \theta)\). Additionally, \( x^2 + y^2 = r^2 \) in polar coordinates.

Express the Equation in Polar Coordinates

Since \( x^2 + y^2 = r^2 \), substitute this into the equation to get \[ r^2 = 16. \] Take the square root of both sides to find \( r \), which gives \( r = 4 \) or \( r = -4 \). However, in polar coordinates, we usually consider \( r = 4 \), as the equation represent a circle centered at the origin with a radius of 4.

Sketch the Graph

Knowing that \( r = 4 \) is a circle centered at the origin, you can sketch it by drawing a circle of radius 4. In polar coordinates, this means every point on the circle is 4 units away from the origin, for every angle \( \theta \).

Key concepts

Rectangular Equation

Rectangular equations are expressions involving the Cartesian coordinates, generally characterized as functions with variables \(x\) and \(y\). Our initial equation is \(x^2 + y^2 = 16\), which classically represents a circle in the Cartesian plane. In this expression, each point on the circle satisfies the equation, indicating that the set of these points forms a perfect circle. Circles in rectangular form center around a fixed point, with this specific example situated at the origin \((0,0)\). The equation entails using square terms, and each component \(x^2\) and \(y^2\) displays symmetry throughout both axes. This highlights the uniform distribution of points all at a consistent distance, demonstrating that the radius is 4 units. When working with rectangular equations, such properties allow simple conversion to polar coordinates by understanding these geometric traits and shifting perspectives.

Coordinate Conversion

Switching between different coordinate systems involves understanding the relationship between various forms. Here, we translate a rectangular equation into polar coordinates. The transformation begins by recalling that in polar form, the relationship of any point \((x, y)\) to \((r, \theta)\) is defined as:

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)
• \(x^2 + y^2 = r^2\)

By using \(x^2 + y^2 = r^2\), this formula directly leads to our polar equation from the given Cartesian equation. Simply substitute and simplify the existing equation. Here, \(x^2 + y^2 = 16\) becomes \(r^2 = 16\), expressing the same circle but now in polar terms.
This conversion gives a new perspective, and while it simplifies the equation to \(r = 4\), it retains all necessary details about the circle's radius and origin-center, preparing it for future applications like graph sketching or analysis.

Circle Equation

The equation \(x^2 + y^2 = 16\) tells us about a circle centered at the origin with a radius of 4. When this translates to polar coordinates, it becomes \(r = 4\). Understanding a circle equation simplifies recognizing attributes such as:

• The center's position (origin in both rectangular and polar coordinates)
• The fixed radius (4 units from the center)
• The concept of symmetry

The polar representation \(r = 4\) gives a complete picture, showing that every point on this circle falls 4 units from the origin, no matter the angle \(\theta\). Unlike rectangular coordinates that need both \(x\) and \(y\) positions to describe a circle, polar equations like \(r = 4\) provide a simpler means to track points with just one variable.
Being simple and elegant, this form aids in efficient graphing and offers clear insights into geometric properties, crucial when sketching or analyzing complex shapes.

Graph Sketching

Sketching a graph effectively communicates geometric ideas, especially when dealing with circles in either rectangular or polar form. In this scenario, transforming \(x^2 + y^2 = 16\) into \(r = 4\) makes the sketch straightforward:

• Identify the origin point as the circle's center.
• Use the radius (4) to draw a perfectly round circle.
• Ensure distance from the origin stays consistent at every angle \(\theta\).

Beginning with the polar approach, every point is plotted by simply marking a 4-unit distance from the origin, regardless of angle, completing a circle beautifully.
Thus, knowing both rectangular and polar forms lays a strong foundation. It equips students to handle graphs effortlessly while building an intuitive grasp of coordinate systems. This practice not only aids in immediate tasks but sharpens skills for more advanced topics later in math studies.