Question

Name the curve with the given polar equation. If it is a conic, give its eccentricity. Sketch the graph. $$ r=4 \sin \theta $$

Short answer

The curve is a circle centered at (0,2) with radius 2.

Step-by-step solution

Identify the Type of Polar Equation

The given equation is in the form \( r = a \, \sin \theta \). This form suggests it might be a circle, as equations of the form \( r = a \, \sin \theta \) or \( r = a \, \cos \theta \) represent circles in polar coordinates.

Determine the Characteristics of the Circle

The equation \( r = 4 \, \sin \theta \) represents a circle centered on the vertical axis (polar axis). In polar coordinates, if \( r = a \, \sin \theta \) where \( a > 0 \), the circle is centered at \( (0, \frac{a}{2}) \) and has a radius of \( \frac{a}{2} \). For this equation, \( a = 4 \), so the circle's center is at \( (0, 2) \) and the radius is 2.

Check for Conic and Eccentricity

Circles can be considered conics with an eccentricity of 0. The formula \( r = 4 \, \sin \theta \) does not have an explicit term for eccentricity like other conics, as it does not fit the standard form for ellipses or hyperbolas, confirming it is a circle.

Sketch the Graph

In polar coordinates, plot the circle with a center at \( (0, 2) \) and radius 2. This circle passes through the origin when \( \theta = \frac{\pi}{2} \), and extends downwards to the line \( \theta = \frac{3\pi}{2} \). It is a complete circle spanning 360 degrees.

Key concepts

Polar Equation

A polar equation is a mathematical expression used to describe the position of a point in polar coordinates. Unlike Cartesian coordinates that use an x and a y value to define the location, polar coordinates use the distance from a fixed point known as the pole and an angle from a fixed direction, typically the positive x-axis. The equation given in the exercise is \( r = 4 \sin \theta \). This tells us that for any angle \( \theta \), the distance from the origin is determined by multiplying 4 by the sine of the angle.

Polar equations can represent various shapes, such as circles, spirals, and roses, depending on their form. In this case, we have an equation similar to \( r = a \sin \theta \), which is a known form for circles in polar coordinates.

Some important points about polar equations:

• They elegantly describe curves that are difficult to express in Cartesian form.
• For the form \( r = a \sin \theta \), if \( a > 0 \), the circle will be shifted upwards along the vertical axis.

Conic Sections

Conic sections are the curves obtained by intersecting a cone with a plane. They include ellipses, parabolas, hyperbolas, and circles. Each conic section has a unique eccentricity (e) that defines its shape.

- **Eccentricity (e):**

• Circular sections have an eccentricity of 0.
• Ellipses (excluding circles) have an eccentricity greater than 0 but less than 1.
• Parabolas have an eccentricity of exactly 1.
• Hyperbolas have an eccentricity greater than 1.

In the context of this exercise, the given polar equation \( r = 4 \sin \theta \) represents a circle with an eccentricity of 0. This is because a circle is a special form of an ellipse where the two foci coincide at the center, making the eccentricity 0. Understanding the eccentricity helps in identifying the nature of the conic section defined by the polar equation.

Circle

A circle in polar coordinates often defined by equations in the form \( r = a \sin \theta \) or \( r = a \cos \theta \). These forms indicate a circle centered along an axis in polar coordinates. For our exercise, we have the equation \( r = 4 \sin \theta \), which is a circle.

Important features of this circle:

• The center is located at \( (0, \frac{a}{2}) \), in this case, at \( (0, 2) \).
• The radius of the circle is \( \frac{a}{2} \), which is 2 for our given \( a = 4 \).
• This circles passes through the origin at \( \theta = \frac{\pi}{2} \) and reaches its maximum when \( \theta = 0 \).

Circles are a fundamental geometric shape and understanding their representation in polar coordinates can be incredibly useful for graphing and geometric interpretations.

Graphing Polar Curves

Graphing polar curves may initially seem challenging, as the approach differs from Cartesian plotting. However, by understanding polar coordinates and practicing different plots, it becomes an intuitive process. In polar plots, each point is determined by an angle and radius.

To graph our exercise's equation \( r = 4 \sin \theta \):

• Determine the maximum value for \( r \). Here, \( r \) varies from 0 to 4 as \( \theta \) changes.
• If \( \theta = \frac{\pi}{2} \), the curve touches the origin.
• The circle centers around \( (0, 2) \) and fully occupies around \( \theta = 0 \) and \( \pi \).

Each point on the curve can be plotted by calculating its radius for specific \( \theta \) values and connecting these plots into a smooth curve. Polar plotting takes practice, but given the nature of polar graphs, visualizing periodic or symmetrical phenomena can become easier than in traditional Cartesian graphs.