Problem 59 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 45-78: Sketch the graph of the polar equation. $$ r=\sqrt{3}-2 \sin \theta $$

Step-by-Step Solution

Answer

The graph sketches a limaçon with a sinusoidal variation due to \( \sin \theta \).

Step 1: Understand the Polar Equation

The given polar equation is \( r = \sqrt{3} - 2 \sin \theta \). In this context, \( r \) represents the radial distance from the origin (pole), and \( \theta \) represents the angle measured from the positive x-axis. The goal is to graph this equation on polar coordinates.

Step 2: Identify Key Features

Converting the equation, note that it involves the sine function which suggests that the graph will have certain symmetry. Given that the maximum value of \( \sin \theta \) is 1 and the minimum is -1, we can calculate the range of \( r \):\( r = \sqrt{3} - 2 \sin \theta \) ranges from \( \sqrt{3} - 2(1) \) to \( \sqrt{3} - 2(-1) \), resulting in \( r \in [\sqrt{3} - 2, \sqrt{3} + 2] \).

Step 3: Determine Important Points

Calculate some specific values to plot: \( \theta = 0 \Rightarrow r = \sqrt{3} \), \( \theta = \frac{\pi}{2} \Rightarrow r = \sqrt{3} - 2 \). Similarly, \( \theta = \pi \Rightarrow r = \sqrt{3} \), and \( \theta = \frac{3\pi}{2} \Rightarrow r = \sqrt{3} + 2 \). By plotting these points, we see the changes in \( r \) as \( \theta \) increases.

Step 4: Plot the Graph

Using the calculated points, sketch the graph. Start by plotting the radial distances from the origin for each of the important angles determined in the previous step. Connect these points knowing that the variation is due to the sinusoidal nature of \( \sin \theta \) resulting in a shape similar to a circle or an ellipse, likely closer to a limaçon due to the behavior of sine.

Step 5: Refine the Sketch with Symmetry

Refine the sketch by considering symmetries and using more points if needed. Due to periodicity and properties of sine, the wave has a symmetric bulge. This involves sketching across different quadrants by considering the inherent symmetric behavior due to \( \sin(-\theta) = -\sin \theta \).

Key Concepts

Polar Coordinates

Polar coordinates provide a different way to represent points in the plane compared to the Cartesian coordinate system. In polar coordinates, each point is defined by a distance from a fixed point known as the origin or pole, and an angle from a fixed direction, usually the positive x-axis.

• The distance is denoted by \( r \), which specifies how far the point is from the origin.
• The angle, \( \theta \), is measured in radians and defines the direction of the point.

Just like every Cartesian point (x, y) corresponds to a unique point in polar coordinates and vice versa. Converting between these systems helps in understanding different graphical shapes, particularly those that are not easily expressed in Cartesian terms. For example, circles and spirals often have simpler expressions in polar coordinates. Understanding this system is crucial when dealing with equations like \( r = \sqrt{3} - 2 \sin \theta \) that become intuitive with angular components.

Polar Equations

Polar equations express relationships between the radial coordinate \( r \) and the angular coordinate \( \theta \). Most notably, they include trigonometric functions which enhance their descriptive power. The equation \( r = \sqrt{3} - 2 \sin \theta \) involves a sinusoidal component indicating that the shape of the graph will have wave-like features.

• Such polar equations can describe classical curves such as limaçons, roses, and spirals.
• The sinusoidal term \( \sin \theta \) introduces periodicity, causing the graph to repeat in each quadrant.

Understanding how these equations work helps to accurately predict the shape and key features of their graphs. By substituting different \( \theta \) values, we can calculate corresponding \( r \) values to plot, unveiling patterns inherent in the polar equation's structure. This forms the basis for graph sketching in polar systems.

Symmetry in Polar Graphs

Symmetry plays a crucial role in simplifying the graphing of polar equations. Understanding symmetries helps reduce the complexity by minimizing the number of points needed to sketch a complete graph. Polar graphs often exhibit symmetries due to the properties of trigonometric functions like sine and cosine.

• If replacing \( \theta \) with \( -\theta \) yields the same equation, the graph is symmetric about the polar axis (line \( \theta = 0 \)).
• If replacing \( r \) with \( -r \) leads to an equivalent equation, symmetry about the origin is present.

Considering the polar equation \( r = \sqrt{3} - 2 \sin \theta \), its symmetry can be analyzed by leveraging properties like \( \sin(-\theta) = -\sin(\theta) \). These insights enable a more refined and less labor-intensive approach to sketching, as symmetrical properties guide the placement of points across the quadrants, leading to a more accurate and quick graph realization.