Problem 49 - Thomas Calculus

Question

Graph the lines and conic sections in Exercises \(47-56\) $$ r=4 \sin \theta $$

Step-by-Step Solution

Answer

The graph is a circle centered at (0, 2) with radius 2.

Step 1: Understand the Equation

The equation given is \( r = 4 \sin \theta \). This is in polar coordinates where \( r \) represents the radius, and \( \theta \) is the angle.

Step 2: Identify the Conic Section

The standard form for polar equations of circles is \( r = a \sin \theta \) or \( r = a \cos \theta \). Since our equation is \( r = 4 \sin \theta \), it represents a circle.

Step 3: Determine Circle's Attributes

For the equation \( r = 4 \sin \theta \), the circle's center is at \( (0, 2) \) in Cartesian coordinates, and its radius is 2.

Step 4: Convert to Cartesian Coordinates

Use the formulas \( x = r \cos \theta \) and \( y = r \sin \theta \). For the equation \( r = 4 \sin \theta \), substituting for \( y \), we get \( r = \frac{4y}{r} \). Simplifying gives the Cartesian equation \( x^2 + y^2 = 4y \).

Step 5: Simplify the Cartesian Equation

The equation \( x^2 + y^2 = 4y \) can be rewritten as \( x^2 + (y-2)^2 = 4 \). This matches the standard form \( (x-h)^2 + (y-k)^2 = r^2 \) for a circle centered at \( (0,2) \) with radius 2.

Step 6: Graph the Circle

Draw a circle centered at point \( (0,2) \) on the Cartesian plane with radius 2. Ensure the circle passes through \( (0,0) \) and \( (0,4) \) touching the x-axis at the origin.

Key Concepts

Conic Sections

Conic sections are fascinating curves formed by the intersection of a plane and a double-napped cone. These sections include circles, ellipses, parabolas, and hyperbolas. What differentiates these curves is the angle and position at which the plane cuts through the cone. Understanding conic sections is fundamental in both geometry and algebra.

• **Circle:** A perfectly round shape that emerges when the cutting plane is parallel to the base of the cone. It is the simplest conic section.
• **Ellipse:** This curve resembles a stretched circle formed when the plane cuts through the cone at an angle.
• **Parabola:** Created when the plane is parallel to the cone's edge. It is a symmetrical open curve.
• **Hyperbola:** Arises when the plane cuts through both napes of the cone. A hyperbola consists of two disconnected curves.

In this example, we work with the equation \( r = 4 \sin \theta \), representing a circle in polar coordinates. This aligns with the standard forms of conic sections, specifically the polar form of a circle.

Cartesian Coordinates

Cartesian coordinates provide a straightforward way to express curves and paths using a grid system, defined by an x-axis (horizontal) and a y-axis (vertical). These coordinates are essential for graphing because they translate complex equations into easily understood visuals. In polar to Cartesian equation conversion, we use the relationships \( x = r \cos \theta \) and \( y = r \sin \theta \).

• **Conversion Process:** Start by substituting the polar equation variables into the Cartesian formulas. For \( r = 4 \sin \theta \), we substitute \( y = r \sin \theta \), simplifying to find the relationship between \( x \) and \( y \).
• **Translated Equation:** The provided circle in polar coordinates \( r = 4 \sin \theta \) converts into the Cartesian form as \( x^2 + y^2 = 4y \).
• **Simplifying:** This equation is simplified to match the standard circle form \( (x-h)^2 + (y-k)^2 = r^2 \), resulting in \( x^2 + (y-2)^2 = 4 \).

This illustrates Cartesian coordinates' power in making the circle's properties more evident, such as its center and radius, in this specific case.

Graphing Circles

Graphing circles is a particularly visual aspect of mathematics and helps many students understand geometric properties intuitively. A circle is defined by its center point and radius.

• **Determine the Center and Radius:** For the equation \( (x-0)^2 + (y-2)^2 = 4 \), the circle's center is \((0, 2)\) and its radius is 2. The form \( (x-h)^2 + (y-k)^2 = r^2 \) is standard for circles, making comparisons straightforward.
• **Sketch the Circle:** Begin by locating the center on the graph. Count outwards from it by the radius, 2 units in every direction, to get an idea of the circle's size.
• **Graph the Circle:** Draw it, ensuring it encompasses areas such as the origin and the point where it meets the line parallel to the x-axis at y = 4, confirming the circle's geometric accuracy.

Using these steps, graphing circles in Cartesian coordinates becomes simpler, aiding in visualizing mathematical relationships effectively.