Problem 103 - Precalculus with Limits

Question

In Exercises 85-108, convert the polar equation to rectangular form. \(r=2\ \sin\ 3\theta\)

Step-by-Step Solution

Answer

The rectangular form of the given polar equation is \(x^2 + y^2 = 36y^2 - 24y^4 + 4y^6/x^2\).

Step 1: Convert the Polar Coordinates

The first step is to rewrite the given equation in terms of \(\sqrt{x^2+y^2}\) and \(y/\sqrt{x^2+y^2}\). This gives us \(\sqrt{x^2+y^2} = 2\ \sin(3\ \arctan(y/x))\)

Step 2: Handle the Multiple of Theta

Since we have a multiple of theta in the sinusoidal function, we use the triple-angle formula for sine, which is defined as \(\sin 3\phi = 3\sin\phi - 4\sin^3\phi\). Thus, to simplify, the given equation becomes: \(\sqrt{x^2+y^2}=2(3(y/\sqrt{x^2+y^2})- 4(y/\sqrt{x^2+y^2})^3)\)

Step 3: Simplify and Make the Conversion

Simplify the equation to convert it entirely into rectangular form: \(\sqrt{x^2+y^2}=6y - 8(y^3/(x^2+y^2))\). Also, squaring both sides to clear the square root gives the final equation: \(x^2 + y^2 = 36y^2 - 24y^4 + 4y^6/x^2\).

Key Concepts

Polar Coordinates

Polar coordinates are a way of representing a point in the plane using two values: the radial distance from a fixed point (usually the origin) and the angle from a fixed direction. Imagine it as a way to describe a point's location using a circle's radius and angle, the way you might aim and shoot an arrow. Here are the key components:

• Radial Distance (r): This is the distance from the origin to the point. It's like the length of a straight line segment drawn from the center to the point.
• Angle (θ): The angle is measured from a fixed direction, typically the positive x-axis, going counterclockwise to the segment connecting the origin to the point.

Polar coordinates are especially useful in scenarios where motion or position naturally involves rotation around a point, like circular motion. They are often used in fields such as engineering, navigation, and physics.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are based on two perpendicular lines, usually known as the x-axis and y-axis. Each point on the plane is represented as ( x, y), where x is the horizontal distance from the origin and y is the vertical distance. Imagine plotting cities on a grid map where you measure distance from the intersection of two roads. Some important aspects include:

• X-coordinate (x): It indicates how far the point is from the origin horizontally. A point with a positive x-coordinate is to the right of the y-axis, while a negative x-coordinate is to the left.
• Y-coordinate (y): This value represents how far the point is vertically from the origin. Positive y-values are above the x-axis, and negative ones are below.

Rectangular coordinates are highly effective for creating graphs or visualizing data in most algebraic problems and analyses. They serve as the fundamental basis for most modern mathematical work.

Triple-Angle Formula

The triple-angle formula is a trigonometric identity used to expand expressions like \( \sin(3\theta) \)'). It helps in converting multiple angle trigonometric equations into simpler components. The key formula to remember is:
\[\sin 3 \phi = 3 \sin \phi - 4 \sin^3 \phi\]
This formula breaks down the complex angle into simpler, more manageable parts using familiar single angle trigonometric components. Here is a quick breakdown:

• The formula comes in handy when simplifying expressions like the one in our original exercise involving sinusoidal functions with multiplied angles.
• Each term in the formula, such as \( 3\sin\phi \), relates back to the basic functions of sine, providing insights into reducing the equation from polar to rectangular form.

The triple-angle formula isn't just for theoretical math but also practical applications, like signal processing where frequencies are analyzed using such trigonometric expansions.