Question

Graph each function in polar coordinates. $$r=\sin 2 \theta-1$$

Short answer

To graph the polar equation \(r=\sin 2\theta - 1\), generate values by selecting key angles, calculate corresponding 'r' for each, plot the points, and sketch the curve. Look for symmetry and looping behavior.

Step-by-step solution

Understand the Polar Coordinate System

In the polar coordinate system, a point on a plane is determined by a distance from a reference point and an angle from a reference direction. The distance from the reference point is denoted as 'r', and the angle is denoted as '\(\theta\)'. A polar equation relates 'r' with '\(\theta\)'.

Generate Values

Create a table of values for '\(\theta\)' and compute the corresponding 'r' values using the polar equation '\(r=\sin 2\theta - 1\)'. It is often useful to choose angles where the sine function has known values, such as 0, \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), \(\frac{\pi}{3}\), \(\frac{\pi}{2}\), and so on.

Plot the Points and Sketch the Graph

Plot the points from your table onto polar graph paper, using the angle '\(\theta\)' and the calculated 'r' value for each point. Connect the points to form the graph of the polar equation. Remember that negative 'r' values mean moving in the opposite direction from the pole.

Identify Symmetry

Examine the graph for symmetry. In this case, since we have a sine function multiplied by 2, we expect the plot to repeat every \(\pi\) radians and be symmetrical about the vertical line (polar axis). This can simplify the plotting process.

Analyze the Graph

After plotting the graph, look for points, where the graph intersects the pole (origin), which occurs when 'r' equals zero. Additionally, check for any loops or intersections that occur, which are common in polar equations involving trigonometric functions.

Key concepts

Polar Coordinate System

The polar coordinate system is a two-dimensional coordinate system where each point on a plane is determined by a distance from a reference point and an angle from a reference direction. Unlike the Cartesian coordinate system, which uses a grid of horizontal and vertical lines to define a point's position through x and y coordinates, the polar system uses a radial distance and an angular coordinate.

In more technical terms, the polar coordinates \( (r, \theta) \) consist of:\begin{itemize}\item A radial coordinate 'r' — which represents the distance from a central point known as the pole (analogous to the origin in Cartesian coordinates).\item An angular coordinate '\theta' (theta) — which represents the counterclockwise angle made from a reference direction, which is typically the positive x-axis in Cartesian coordinates, called the polar axis.\end{itemize}
Students often encounter polar coordinates in problems involving trigonometry, calculus, and complex numbers. It's crucial to understand that negative values of 'r' simply indicate that the point is in the direction opposite to what the angle 'theta' normally suggests.

Polar Equations

Polar equations express the relationship between the radius 'r' and the angle '\theta'. They can describe curves or shapes that would be more complex or impossible to represent with Cartesian equations. For instance, circles, spirals, and rose curves, which have natural circular or radial symmetries, are much more easily described in polar coordinates.

A standard form of a polar equation is \( r = f(\theta) \), where \( f \) is any function that computes a radius for each angle '\theta'. Since trigonometric functions naturally relate angles to ratios of side lengths in right triangles, they appear frequently in polar equations. They can produce beautifully symmetrical patterns when graphed. For example, the equation from our exercise, \( r = \sin(2\theta) - 1 \) is a polar equation where the radius depends on the sine of twice the angle minus one. Understanding how the radius changes with the angle is key to plotting these equations accurately.

Plotting Polar Functions

Plotting polar functions involves translating the polar equation into a set of points and then sketching the graph. This can be done by creating a table of values where you pick several values of '\theta', calculate the corresponding 'r' for each one using the polar equation, and then plot these \( (r, \theta) \) pairs on the polar graph.

Polar graph paper is helpful because it typically has concentric circles representing values of 'r' and radiating lines for angles '\theta'. Here are some steps to follow when plotting polar functions:\begin{itemize}\item Begin by choosing values for '\theta' that span the range of the function, especially looking for values where trigonometric functions reach their maximums, minimums, or zeroes.\item For each selected value of '\theta', calculate the corresponding 'r'.\item Plot the point by moving 'r' units away from the pole at the angle '\theta'.\item Connect the points smoothly to reveal the graph of the function.\end{itemize}
As a general tip, symmetry can also be used to simplify the plotting process - if a function is symmetrical about an axis or a point, you may only need to plot half of the graph.

Trigonometric Functions in Polar Coordinates

Trigonometric functions such as sine, cosine, and tangent have interesting interpretations when applied in polar coordinates. They can be used to generate various polar graphs including circles, limaçons, lemniscates, and rose curves, each with distinct characteristics and symmetries.

In the polar equation \( r = \sin(2\theta) - 1 \) from our exercise, the sine function determines the radius for each angle. Since the sine function has a known period of \( 2\pi \) radians, multiplying the angle by 2 before applying the sine function modifies this period. This results in the graph repeating every \( \pi \) radians instead. When graphing, this adjusted period means we can expect certain symmetrical features over specified intervals, which can greatly aid in the drawing process. By recognizing these patterns and how the trigonometric function shapes the graph, students can not only plot the points precisely but also gain a deeper understanding of the polar graphs generated by various trigonometric functions.