Problem 21 - Precalculus Mathematics for Calculus

Question

Sketch a graph of the polar equation, and express the equation in rectangular coordinates. $$r=6 \sin \theta$$

Step-by-Step Solution

Answer

Rectangular form: \( x^2 + y^2 = 6y \) (Circle passing through origin on the positive y-axis).

Step 1: Understand the Polar Equation

The given polar equation is \( r = 6 \sin \theta \). This equation describes a relationship between the polar coordinates \( r \) (the radius) and \( \theta \) (the angle).

Step 2: Identify the relevant trigonometric identities

Based on the given expression or equation, identify which trigonometric identities (Pythagorean, double-angle, sum/difference, etc.) are applicable.

Step 3: Apply the identities and simplify

Apply the identified identities to transform the expression. Simplify step by step, combining like terms and reducing fractions where possible.

Step 4: Solve or evaluate

If solving an equation, isolate the trigonometric function and find the angle(s). If evaluating, compute the final numerical value.

Step 5: State the result

Express the final answer, including all solutions in the required domain if solving an equation.

Step 6: Conclude with the answer

Rectangular form: \( x^2 + y^2 = 6y \) (Circle passing through origin on the positive y-axis).

Key Concepts

Polar to Rectangular Coordinates Conversion

Polar coordinates \((r, \theta)\) are a way to represent points in a plane using a distance from the origin \(r\), and an angle \(\theta\) from the positive x-axis. Rectangular coordinates, or Cartesian coordinates, use a pair \((x, y)\) to pinpoint a location in the plane.

To convert from polar to rectangular coordinates, we use the following equations:

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)

For the polar equation \(r = 6 \sin \theta\), we substitute \(r\) and \(\theta\) using the trigonometric identities for \(x\) and \(y\). We know:

• \(y = 6 \sin \theta\)

Substitute back \(r = y\), we get that the rectangular equation is: \(x^2 + y^2 = y^2\). So the conversion gives us a familiar form, a circle equation, when we simplify: \((x^2 + y^2) = 6y\). This characterizes the shape described in a Cartesian coordinate.Hence, converting between these coordinate systems involves using trigonometric relationships to express a polar equation in an easily understandable rectangular form.

Graphing Polar Equations

Graphing polar equations involves plotting points based on their distance and angle from a central point. The equation \(r = 6 \sin \theta\) tells us a few key things about the graph we're going to sketch.Since \(r\) is always non-negative and assuming \(\theta\) ranges from 0 to \(2\pi\), we understand that \(r = 6 \sin \theta\) corresponds to values of \(r\) from 0 to 6 as \(\sin \theta\) oscillates.

• At \(\theta = \frac{\pi}{2}\), \(\sin \theta = 1\) and \(r = 6\), reaching its peak distance.
• At \(\theta = 0\) or \(\pi\), \(\sin \theta = 0\) so \(r\) returns to 0.

This polar equation typically forms a circle centered at \((0,3)\) in rectangular coordinates. Graphing it can be simplified by recognizing its sinusoidal symmetry, allowing us to plot a few critical points and trace a smooth line between them.
Visualizing polar equations demands familiarity with their unique forms and sketching them promotes an understanding of the impact of coefficients and angles, enriching one’s grasp of polar coordinate systems.

Trigonometric Functions in Polar Coordinates

Trigonometric functions play a fundamental role in polar coordinates, specifically when understanding how angles and distances relate to each other. The polar equation \(r = 6 \sin \theta\) heavily relies on the sine function to define radius based on the angle.In polar systems, trigonometric functions serve to maintain circular symmetry and reveal periodic patterns. The sine function, \(\sin \theta\), varies between -1 and 1, influencing the perceived radius \(r\) accordingly:

• When \(\sin \theta\) is positive, \(r\) stretches outwards, forming part of a circle.
• When \(\sin \theta\) is zero, \(r = 0\), producing the loop or returning to the origin.

These functions are crucial for converting into and interpreting graphs and values within the coordinate system. They help to understand how quickly or slowly \(r\) changes with \(\theta\), revealing the structure or shape outlined by a polar equation.Mastering trigonometric functions within this framework empowers students to better discern their foundational role in both plotting polar equations and transitioning between different coordinate systems.