Question

Write the polar equation \(3 r=\sin \theta\) as an equation in rectangular coordinates. Identify the equation and graph it.

Short answer

\[ 3x^2 + 3y^2 - y = 0 \] is a circle.

Step-by-step solution

Identify Polar Equation

Given polar equation: \[ 3r = \sin \theta \]

Express Variables

Express the variables in the polar equation in terms of rectangular coordinates. Recall that: \[ x = r \cos \theta \] \[ y = r \sin \theta \] and \[ r = \sqrt{x^2 + y^2} \]

Express \sin \theta

Since \sin \theta = \frac{y}{r}, rewrite the given equation \[ 3r = \sin \theta \] as \[ 3r = \frac{y}{r} \]

Isolate r

Isolate r by multiplying both sides by r: \[ 3r^2 = y \]

Substitute for r

Substitute \sqrt{x^2 + y^2} for r: \[ 3(x^2 + y^2) = y \]

Distribute and Rearrange

Distribute 3 and rearrange the equation to standard form: \[ 3x^2 + 3y^2 - y = 0 \]

Identify the Equation

Identify the resulting equation. This equation represents a conic section, specifically a circle.

Graph the Equation

Plot the graph of the equation \[ 3x^2 + 3y^2 - y = 0 \] This is a circle centered at the origin, but shifted slightly upwards.

Key concepts

Polar Coordinates

In polar coordinates, each point in a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (origin) is often denoted as \(O\), and the reference direction is usually the positive x-axis. The coordinates are given as \( (r, \theta) \), where:

• \(r\): the radial distance from the origin to the point
• \(\theta\): the angle measured counterclockwise from the positive x-axis to the line segment connecting the origin to the point

For example, in the given exercise, the polar equation is \(3r = \sin \theta\). By understanding polar coordinates, we focus on the distance and angle measurements.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, use two perpendicular axes: the x-axis and the y-axis. Any point in the plane is represented as \( (x, y) \), where:

• \(x\): the horizontal distance from the origin
• \(y\): the vertical distance from the origin

Connecting these to polar coordinates, we have useful conversion formulas:

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)
• \(r = \sqrt{x^2 + y^2}\)

Understanding rectangular coordinates helps in converting between different coordinate systems, as seen when turning \(3r = \sin \theta\) into \(3(x^2 + y^2) = y\).

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the angle involved. They are crucial in converting between coordinate systems. In particular:

• \(\sin \theta = \frac{y}{r}\)

On transforming the exercise equation, we use \(\sin \theta = \frac{y}{r}\) to rewrite the polar equation \(3r = \sin \theta\) as \(3r = \frac{y}{r}\), simplifying it into a form that involves \(r\), which is then replaced by \(\r = \sqrt{x^2 + y^2}\).

Conic Sections

Conic sections are curves obtained by slicing a cone with a plane. They include circles, ellipses, parabolas, and hyperbolas. The equation \(3(x^2 + y^2) = y\) is a form of a conic section. Specifically, simplifying and analyzing it reveals that it represents a circle:

• The general form, \(Ax^2 + By^2 + Cy + D = 0\), identifies different conics.
• In our case, \(3x^2 + 3y^2 - y = 0\) is a circle centered at the origin but slightly shifted due to the y-term.
• Drawing its graph, one can see the symmetric circular shape.

Understanding conic sections is essential for recognizing and graphing the resultant form from coordinate conversions.