Problem 46 - Algebra and Trigonometry

Question

A polar equation is given. (a) Express the polar equation in parametric form. (b) Use a graphing device to graph the parametric equations you found in part (a). $$ r=2^{\sin \theta} $$

Step-by-Step Solution

Answer

Parametric form: \(x(t) = 2^{\sin t} \cos t\), \(y(t) = 2^{\sin t} \sin t\). Graph using \( t \) as the parameter.

Step 1: Review Polar Equation and Identify Parameters

The given polar equation is \( r = 2^{\sin \theta} \), where \( \theta \) represents the angle in the polar coordinate system. We need to convert this polar equation into parametric form.

Step 2: Connect Polar to Cartesian Coordinates

Remember that the relationship between polar coordinates \( (r, \theta) \) and Cartesian coordinates \( (x, y) \) is \( x = r \cos \theta \) and \( y = r \sin \theta \). In parametric form, \( \theta \) becomes the parameter \( t \).

Step 3: Express the Parametric Equations

Substitute \( r = 2^{\sin t} \) into the Cartesian coordinate formulas to get:\[x = 2^{\sin t} \cos t\]\[y = 2^{\sin t} \sin t\]These equations express the given polar equation in parametric form with \( t \) as the parameter.

Step 4: Graph the Parametric Equations

Using a graphing device, like Desmos or Graphing Calculator, plot the parametric equations:\[x(t) = 2^{\sin t} \cos t\]\[y(t) = 2^{\sin t} \sin t\]For \( t \) in the interval \( [0, 2\pi] \) or multiples thereof to see the full cycle of the graph. Observe the shape and symmetry of the graph to understand the behavior of the equation.

Key Concepts

Polar Coordinates

Polar coordinates provide a unique way to describe a point in the plane using a distance and an angle. Instead of using the traditional Cartesian coordinates (x, y), polar coordinates use a radius, denoted as \( r \), and an angle, denoted as \( \theta \). This system is particularly useful for dealing with problems involving circular or rotational symmetry.

To illustrate, a point in the polar system is defined as \((r, \theta)\). Here, \( r \) represents the distance from the origin (or center, often called the pole), while \( \theta \) is the angle from the positive x-axis, measured in a counterclockwise direction.

One of the main advantages of polar coordinates is their ability to easily handle problems involving circular paths. For example, the simple equation \( r = 1 \) represents a circle with a radius of 1, centered at the origin. In such a format, complicated functions involving sine and cosine can be expressed and visualized more naturally than in the Cartesian format.

Graphing Parametric Equations

Graphing parametric equations involves plotting curves defined by a set of functions. In our case, the converted parametric form from the polar equation is:\[x(t) = 2^{\sin t} \cos t\]\[y(t) = 2^{\sin t} \sin t\]
These equations define the x and y coordinates in terms of a third variable, typically denoted as \( t \), representing a parameter.

To visualize these curves, use tools like graphing calculators or software such as Desmos. By plugging different values of \( t \) into the equations within a selected interval (commonly \([0, 2\pi]\)), you can observe how the curve forms over that domain.

• Parametric plots can display the motion and dynamics of a point as it draws a path in the plane, giving insights into the behavior of the underlying function.
• They are especially useful in applications involving motion, physics, or anything related to cyclical patterns or oscillations.

The ability to see the relationship between \( x \), \( y \), and \( t \) offers a deeper understanding and can reveal fascinating symmetrical properties or unique patterns like spirals, circles, or other shapes.

Polar to Cartesian Conversion

Converting between polar and Cartesian coordinates requires understanding the relationship between the two systems. The conversion is facilitated by the relationships:

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

These equations allow you to transform a point from polar \((r, \theta)\) to its equivalent Cartesian coordinates \((x, y)\). This transformation is crucial when we want to make use of polar representations within a Cartesian framework, where many computational tools and methods are established.

In our specific example, to convert the polar equation \( r = 2^{\sin \theta} \) into Cartesian coordinates, we use the parametric form as an intermediary. By letting \( \theta = t \) and applying the formulas above, we derive the equations for \( x(t) \) and \( y(t) \). This step is pivotal as it bridges the two coordinate systems seamlessly.

The conversion is not merely procedural but opens doors to analyzing and solving geometric problems in a context that best suits the problem's nature.