Problem 75 - Precalculus: Functions and Graphs

Question

Sketch the graph of the polar equation. $$r=\sqrt{3}-2 \sin \theta$$

Step-by-Step Solution

Answer

The polar equation sketches as a circle intersecting specific angles, centered vertically.

Step 1: Convert to Cartesian Coordinates

To sketch the polar equation, we could start from its Cartesian form. The polar equations are given as \( r = \sqrt{3} - 2 \sin \theta \). We use the relationships \( x = r \cos \theta \) and \( y = r \sin \theta \). This equation can be rearranged as \( r = \frac{\sqrt{3}}{1 \pm 2\sin\theta} \), but let's add some analysis steps first.

Step 2: Analyze the Parameters

By letting \( \theta = \frac{\pi}{2} \), we find \( r = \sqrt{3} - 2 \), leading to \( r = -1 \). Likewise, for \( \theta = 0 \), \( r = \sqrt{3} \). This implies the graph will be a circle in the plane with specific focus points.

Step 3: Identify Key Points of the Circle

Set \( r = 0 \) to find intercepts. Thus, \( \sqrt{3} - 2 \sin \theta = 0 \) giving \( \sin \theta = \frac{\sqrt{3}}{2} \). This occurs when \( \theta = \frac{\pi}{3}, \frac{2\pi}{3} \). These points translate to \((0, \theta)\) in polar coordinates, indicating key graph points at these angles.

Step 4: Define the Circle's Characteristics

Determine the nature and center of this equation. The equation is of a circle offset vertically. There is one circle represented and it loops around with certain key features.

Step 5: Graph the Circle

Given the intersections and parameters, the graph represents a vertical loop intersecting at \( 2\sin\theta = \sqrt{3} \), forming a circle with position centered at \( (0,0) \) radially spanning the space in an unconventional layout, with specific focus at two angles.

Key Concepts

Polar Equations

Polar equations describe curves using the polar coordinate system, where each point on a plane is determined by a distance from a reference point and an angle from a reference direction. This coordinate system is particularly useful for plotting circles and spirals. The polar equation given in the exercise is \( r = \sqrt{3} - 2 \sin \theta \). Here, \( r \) represents the radius or the distance from the origin, and \( \theta \) is the angle measured in radians.

• In polar equations, changes in \( \theta \) can cause the radius \( r \) to shift, vary, or even become negative.
• In our equation, as \( \theta \) varies, the term \(-2 \sin \theta\) affects the radius, thereby altering the shape and size of the graph.

Understanding polar equations is crucial when analyzing curves that are more naturally expressed in polar form, such as circles and lemniscates.

Cartesian Coordinates

In contrast to polar coordinates, Cartesian coordinates use two values to determine a point's location in a plane: \( x \) and \( y \). These relate directly to distances along vertical and horizontal axes. Converting a polar equation to Cartesian form allows for easier visualization and analysis using tools familiar from basic geometry.

To translate a polar equation, like \( r = \sqrt{3} - 2 \sin \theta \), into Cartesian coordinates, you use the relationships:

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

From these, the circle analysis becomes more manageable, showing how expressions in polar form convert to equations of circles or lines in Cartesian space. Understanding these conversions provides deeper insight into the mathematical properties of shapes using different coordinate systems that often simplify problem-solving.

Polar Graphing

Polar graphing is the art of mapping a polar equation onto a plane, showcasing the unique paths and forms that these equations create. In polar graphs, curves derive their beauty from how the angle \( \theta \) and radius \( r \) interact. Points are plotted using \( r \) as the distance from the origin and \( \theta \) as the rotational position. Wonderfully diverse curves, including circles, roses, and spirals, emerge from this system.

To graph the curve \( r = \sqrt{3} - 2 \sin \theta \), we observe several properties:

• The graph forms a circle with its size and position dependent on the sine function component.
• Key points appear where \( \, \sin \theta\) involves critical angles like \( \theta = \frac{\pi}{3} \) and \( \theta = \frac{2\pi}{3} \), aligning with the symmetry of sine's periodic pattern.

This visualization method makes it easier to find and interpret characteristics such as intercepts, loops, or lobes present in the graphs of polar equations.

Circle Analysis

Circle analysis in polar coordinates involves understanding the relationship between radius and angle to determine the circle's size, position, and orientation. With the given polar equation \( r = \sqrt{3} - 2 \sin \theta \), the analysis identifies this shape's characteristics:

• The circle is offset and vertically loops around the origin.
• Its intersection points or key positions are at specific values of \( \theta \), where the sine function drives \( r \) to zero.
• These angles correspond to the sine function's critical values, leading to circle points like \( \theta = \frac{\pi}{3} \) and \( \theta = \frac{2\pi}{3} \).

Understanding these properties helps interpret how the graph will behave visually and geometrically. Circle analysis ensures that we grasp how various polar equations form circular shapes and how these translate into intricate or familiar designs.