Problem 73 - Precalculus with Limits

Question

In Exercises 65-84, convert the rectangular equation to polar form. Assume \(a>0\). \(3x-y+2=0\)

Step-by-Step Solution

Answer

The equation in polar form is \(r = -2 / (3 \cos(\theta) - \sin(\theta))\).

Step 1: Substitute Cartesian coordinates with Polar coordinates

For the given equation \(3x - y + 2 = 0\), let's substitute \(x = r \cos (\theta)\) and \(y = r \sin (\theta)\). We get: \(3r \cos (\theta) - r \sin (\theta) + 2 = 0\)

Step 2: Factor out r

Rearrange the equation and factor out \(r\) to better represent it in polar form: \(r (3 \cos(\theta) - \sin(\theta)) = -2\)

Step 3: Complete the conversion

Finally, divide through by the trigonometric terms to obtain \(r\) alone on one side of the equation, completing the conversion to polar form: \(r = -2 / (3 \cos(\theta) - \sin(\theta))\)

Step 4: Check the final equation

Make sure the final equation does not violate the assumption \(a>0\) and hence \(r>0\). In this case, it doesn't, so the conversion is complete.

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a system where a point in a plane is identified by a pair of numbers. These numbers represent the "x" and "y" positions relative to two intersecting perpendicular lines called axes. The horizontal axis is the x-axis, and the vertical one is the y-axis.

For any point, the pair \(x, y\) describes its location by determining its distances from these two axes. In two-dimensional space, this is a very common method to specify the position of points.

• "x" is the horizontal distance from the vertical axis (y-axis).
• "y" is the vertical distance from the horizontal axis (x-axis).

Imagine you are playing a board game where each square on the board represents a location. The position of each square can be described by these coordinates, just like coordinates in math describe a location on the plane.

Trigonometric Form

The trigonometric form of coordinates is often used when working with angles and radii, such as in polar coordinates. In this system, a point is determined by the distance from a central point (the pole) and an angle measured from a fixed direction (usually the positive x-axis).

To convert rectangular coordinates to this form, we use trigonometric identities:

• The formula \(x = r \cos(\theta)\) relates the x-coordinate to the distance from the origin ("r"), the angle \(\theta\), and the cosine function.
• Similarly, \(y = r \sin(\theta)\) relates the y-coordinate to the radius, the angle, and the sine function.

This method allows us to express points using these trigonometric functions, which can simplify many mathematical problems, especially those involving rotation or periodic movements.

Coordinate Transformation

Coordinate transformation is a process of changing the description of a point's position from one coordinate system to another. This is especially useful when solving problems where certain operations are more easily handled in a specific coordinate system.

For example, converting a rectangular coordinate system to a polar one can be beneficial for solving equations that have rotational symmetry or require angle measurements. This involves:

• Identifying the relationship between the x and y coordinates and polar coordinates through trigonometric concepts (like replacing \(x\) with \(r\cos(\theta)\) and \(y\) with \(r\sin(\theta)\)).
• Simplifying the equation to highlight how distance and angles are interacting in the given problem, as seen in the conversion steps where a rectangular equation is expressed in terms of "r" and "\(\theta\)".

In our example, this transformation helps solve the equation with respect to the conditions provided, like ensuring positive values for the distance (\(r\)) when converting equations between different coordinate systems.