Problem 163 - Calculus Volume 3

Question

Convert the polar equation to rectangular form and sketch its graph. $$ r=4 \sin \theta $$

Step-by-Step Solution

Answer

The rectangular form is \(x^2 + (y-2)^2 = 4\), a circle centered at (0, 2) with radius 2.

Step 1: Convert Polar to Rectangular Coordinates - Step 1

Square both sides of the given polar equation. We have the equation: \(r = 4 \sin \theta\). Square the equation to get \(r^2 = (4 \sin \theta)^2 = 16 \sin^2 \theta\).

Step 2: Express in Rectangular Variables - Step 2

Use the identities: \(x = r \cos \theta\), \(y = r \sin \theta\), and \(r^2 = x^2 + y^2\). Substitute \(\sin \theta = \frac{y}{r}\) into the equation: \(16 \sin^2 \theta = 16 \left(\frac{y}{r}\right)^2\). This can be written as \(16 \cdot \frac{y^2}{r^2}\) since \(\sin^2 \theta = \left(\frac{y}{r}\right)^2\).

Step 3: Combine and Simplify - Step 3

Using \(r^2 = x^2 + y^2\), substitute for \(r^2\) into the equation: \(x^2 + y^2 = 16 \cdot \frac{y^2}{x^2 + y^2}\). Multiply both sides by \(x^2 + y^2\) to eliminate the fraction: \((x^2 + y^2)^2 = 16y^2\).

Step 4: Final Rectangular Equation - Step 4

Expand \((x^2 + y^2)^2 = 16y^2\) to get \(x^2 + y^2 = 4y\). Rearrange the equation to its standard form: \(x^2 + (y - 2)^2 = 4\). This is the equation of a circle with center at \((0,2)\) and radius 2.

Step 5: Sketch the Graph - Step 5

Plot the center of the circle at \((0,2)\) on the Cartesian plane. Draw a circle with radius 2 units from the center. This is the graphical representation of the rectangular equation derived from the original polar equation.

Key Concepts

Polar Coordinates

Polar coordinates provide a way of describing a point in a plane using a radius and an angle. In this system, each point is determined by:

• Radius (\(r\)): The distance from the origin (also known as the pole).
• Angle (\(\theta\)): The angle measured from the positive x-axis to the line connecting the point to the origin.

This system is particularly useful for dealing with problems that have circular or rotational symmetry.
For example, the polar equation \(r = 4 \sin \theta\) can represent a curve where the distance from the origin varies according to the angle. Understanding polar coordinates provides clarity in transforming and representing geometric shapes in various forms.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, consist of ordered pairs \((x, y)\) that specify the location of a point in a plane by its horizontal and vertical distances from the origin. The coordinates are derived using:

• \(x = r \cos \theta\) - the horizontal projection on the x-axis.
• \(y = r \sin \theta\) - the vertical projection on the y-axis.

This way of representing points is often more intuitive for understanding spatial relationships, especially in algebra.
In the exercise, we began with polar coordinates and used trigonometric identities to express them in rectangular form, ultimately making it simpler to sketch and interpret the graph on a Cartesian plane.

Trigonometric Identities

Trigonometric identities are fundamental tools in mathematics that relate the angles and sides of a triangle, extending their application to complex shapes and rotations.

• A basic identity we use is \(\sin \theta = \frac{y}{r}\)
• Another, derived from Pythagoras’ theorem, is \(r^2 = x^2 + y^2\).

These identities help in converting coordinates from one system to another. In this exercise, we expressed \(\sin \theta\) in terms of \(y\) and \(r\), enabling us to eliminate trigonometric terms and focus solely on \(x\) and \(y\).
By manipulating these identities, we converted the polar function \(r = 4 \sin \theta\) into the rectangular form \(x^2 + (y - 2)^2 = 4\), revealing the circular nature of the curve in a more intuitive way.

Equation of a Circle

The equation of a circle in rectangular coordinates typically appears in the format \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.
In this example, after converting the polar equation \(r = 4 \sin \theta\), we derived the rectangular equation \(x^2 + (y - 2)^2 = 4\).

• Center: \((0, 2)\)
• Radius: \(2\)

This equation clearly indicates a circle centered at \((0, 2)\) with a radius of 2 on the Cartesian plane.
Understanding how to derive and interpret this formula helps in visualizing geometric figures and solving more advanced mathematical problems. It connects the polar representation of shapes into a form that is easily graphed and comprehended in a spatial context.