Problem 51 - Algebra and Trigonometry Real Mathematics, Real People

Question

Convert the rectangular equation to polar form. Assume \(a<0\) $$y=a$$

Step-by-Step Solution

Answer

The polar form of the equation \(y = a\), with \(a < 0\) is \(r = -a\).

Step 1: Understanding the Direction

The given equation is \(y=a\), a parallel line to the x-axis that runs through the point (0, a). Because a is negative \(y=a\) will be below the origin, which means that in a polar coordinate system the angle θ would be \(\frac{3\pi}{2}\) radians or 270°.

Step 2: Conversion to Polar Coordinates

To convert this equation into the polar form we need to use the relation \(y=r\sin(\theta)\) which is the conversion formula from rectangular to polar coordinates for the y-coordinate. Substituting y by \(r\sin(\theta)\), the equation becomes \(r\sin(\theta) = a\). Division by a gives \(r = \frac{a}{\sin(\theta)}\). Considering the direction, we have \(\theta = \frac{3\pi}{2}\), which gives \(r = \frac{a}{\sin(\frac{3\pi}{2})}\). Because \(\sin(\frac{3\pi}{2}) = -1\), then the polar form is \(r = -a\).

Key Concepts

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, is one of the most widely used systems for representing points in a plane. Each point is determined by an ordered pair of numbers \( (x, y) \) that stand for its horizontal (x-coordinate) and vertical positions (y-coordinate) relative to two perpendicular lines called axes, namely the x-axis and the y-axis. The point where these axes intersect is called the origin, denoted as \( (0, 0) \).

In the context of the given exercise, the equation \( y = a \) represents a horizontal line that is parallel to the x-axis. The negative value of \( a \) indicates that this line is below the x-axis. Understanding how the line is positioned in a rectangular coordinate system is crucial when converting it into its polar counterpart.

Polar Coordinate System

Unlike the rectangular system, the polar coordinate system represents points in a plane using a distance and an angle. Each point is defined by a radial distance \( r \) from the origin and an angle \( θ \) measured from the positive x-axis, also referred to as the polar axis. Distance \( r \) is always non-negative, while angle \( θ \) is typically measured in radians and can range between \( 0 \) and \( 2π \) radians, or \( 0^\circ \) to \( 360^\circ \) in degrees.

For our exercise, converting the rectangular equation \( y = a \) into polar form required understanding that when \( y \) is negative, \( θ \) is \( \frac{3π}{2} \) radians, because angles in polar coordinates are measured counterclockwise from the positive x-axis. The polar coordinates represent not just a position, but also a direction of approach to that position.

Sine Function in Polar Coordinates

The sine function plays a key role in translating between rectangular and polar coordinates. It describes the relationship between the y-coordinate of a point in rectangular coordinates and the angle \( θ \) and radius \( r \) in polar coordinates using the formula \( y = r \sin(θ) \).

In our example, using this relationship allowed the conversion of the equation \( y = a \) into polar form. By replacing \( y \) with \( r \sin(θ) \) and considering that for \( θ = \frac{3π}{2} \) radians, the sine function yields \( \sin(\frac{3π}{2}) = -1 \) — this helped us arrive at the polar equation \( r = -a \) when \( a < 0 \). The sine function helps visualize how a point's height (y-coordinate) is related to its position on the unit circle, further solidifying one's understanding of how the polar coordinate system captures the essence of a point's location.