Problem 33 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The graph is a limaçon with an inner loop, symmetrical about the horizontal axis.

Step 1: Understand the Polar Equation

The given polar equation is \( r = \sqrt{3} - 2 \sin \theta \). This equation expresses \( r \), the radial distance, as a function of \( \theta \), the angle.

Step 2: Recognize the Type of Curve

The equation \( r = a - b \sin \theta \) represents a limaçon. In this case, \( a = \sqrt{3} \) and \( b = 2 \). Because \( b > a \), the limaçon will have an inner loop.

Step 3: Determine Key Points

To find key points, consider specific values of \( \theta \):- For \( \theta = \frac{\pi}{2} \), \( r = \sqrt{3} - 2(1) = \sqrt{3} - 2 \).- For \( \theta = \frac{3\pi}{2} \), \( r = \sqrt{3} - 2(-1) = \sqrt{3} + 2 \).- For \( \theta = 0 \), \( r = \sqrt{3} \).- For \( \theta = \pi \), \( r = \sqrt{3} \).

Step 4: Find the Theta for Maximum and Minimum R

The maximum radius occurs when \( \sin \theta = -1 \), yielding \( r_{max} = \sqrt{3} + 2 \). The minimum (negative) radius occurs when \( \sin \theta = 1 \), giving \( r_{min} = \sqrt{3} - 2 \).

Step 5: Plot the Key Points on Polar Coordinate System

Plot the points on the polar coordinate plane for the calculated values. Remember that for a negative \( r \) value, plot the equivalent positive value at \( \theta + \pi \).For example, if \( r = \sqrt{3} - 2 \) is negative, plot at \( \theta = \frac{\pi}{2} + \pi = \frac{3\pi}{2} \).

Step 6: Draw the Limaçon with Inner Loop

Using the calculated points and understanding that \( b > a \), sketch a limaçon shape that includes an inner loop starting at the origin when \( r = 0 \) (found when \( \sin \theta = \frac{rac{\sqrt{3}}{2}}{1} \) or similarly calculated). Ensure the sketch reflects symmetry about the horizontal line passing through the pole.

Key Concepts

Limaçon

A Limaçon is a type of polar graph characterized by distinctive loops and heart-like shapes. This curve is usually defined by equations of the form \(r = a \, \pm \, b \sin \theta\) or \(r = a \, \pm \, b \cos \theta\). Depending on the relation between \(a\) and \(b\), the Limaçon can have different appearances:

• If \(b > a\), the Limaçon will have an inner loop, as is the case with the given equation \(r = \sqrt{3} - 2 \sin \theta\).
• If \(b = a\), the Limaçon forms a cardioid, which resembles the shape of a heart.
• If \(b < a\), the Limaçon will not have a loop, resembling more of a distorted circle.

The presence of the inner loop is determined by the specific values of \(a\) and \(b\). In this exercise, since \(b = 2\) is greater than \(a = \sqrt{3}\), it results in an interesting looped structure on the graph that starts at the origin for certain angles.

Polar Coordinates

Unlike traditional Cartesian coordinates that use \(x\) and \(y\) to locate points in a plane, polar coordinates utilize the distance from the origin \(r\) (the radial distance) and the angle \(\theta\) from a reference direction. This system is especially useful for plotting curves like Limaçons, which naturally have radial symmetry.

• The angle \(\theta\) is often measured in radians, moving counter-clockwise from the positive x-axis direction.
• The radial coordinate \(r\) can be positive or negative; when negative, the point is located in the opposite direction, effectively adding \(\pi\) to \(\theta\).

To sketch the given polar equation \(r = \sqrt{3} - 2 \sin \theta\), key values of \(\theta\) such as \(0, \frac{\pi}{2}, \pi,\) and \(\frac{3\pi}{2}\) are considered to calculate corresponding \(r\) values. This reveals essential points for drawing the graph on a polar grid.

Trigonometric Functions

Trigonometric functions like sine and cosine play a fundamental role in polar equations, as they influence the radial distance \(r\) for given angles \(\theta\). For the equation \(r = \sqrt{3} - 2 \sin \theta\):

• \(\sin \theta\) oscillates between -1 and 1, affecting \(r\) considerably. When \(\sin \theta = 1\), \(r\) achieves its minimum value, and when \(\sin \theta = -1\), \(r\) reaches its maximum.
• These functions introduce periodic behavior, which is pivotal when graphing the Limaçon since they determine the curve's oscillatory nature around the radial directions.

In this graph, the sine function establishes where the inner loop appears and helps identify the maximum and minimum radial distances, crucial for accurately sketching the polar graph's unique shape.