Problem 84 - Algebra and Trigonometry

Question

The letters \(x\) and \(y\) represent rectangular coordinates. Write each equation using polar coordinates \((r, \theta) .\) $$ r=4 $$

Step-by-Step Solution

Answer

The polar coordinate equation is \(r = 4\).

Step 1: Identify the given equation

The equation given is in rectangular coordinates: \[ r = 4 \]

Step 2: Convert to polar coordinates

In polar coordinates, the variable \(r\) represents the distance from the origin to the point, which is already given as 4. Therefore, the equation in polar coordinates is \(r = 4\).

Step 3: Verify the conversion

Since the given rectangular equation was \( r = 4 \) and \(r\) is already used in polar coordinates to denote the radius, this confirms the equation in polar form is accurately represented as \(r = 4\).

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a way to specify the location of a point on a plane using two values. These values are typically labeled as \(x,y\).

The \(x\) coordinate represents the horizontal distance from the origin, while the \(y\) coordinate indicates the vertical distance.
For example, the point \(3, 4\) is 3 units to the right of the origin and 4 units up.

Rectangular coordinates are particularly useful in problems involving straight lines, slopes, and functions. They work well for describing objects that fit nicely into a grid.
For this reason, they are widely used in algebra and calculus.

Polar Coordinates

Polar coordinates provide an alternative way to define the location of a point on a plane. Instead of using horizontal and vertical distances, polar coordinates use a radius and an angle.

The radius (r) is the distance from the origin to the point, while the angle (\(\theta\)) measures rotation from the positive x-axis.
For example, the coordinates \(4, \pi/2\) represent a point that is 4 units away from the origin and rotated 90 degrees from the positive x-axis.

Polar coordinates are very useful when dealing with circular and spiral patterns, as well as when solving problems involving angles and distances from a central point.

Coordinate Conversion

Coordinate conversion is the process of changing coordinates from one system to another, such as from rectangular to polar coordinates.

To convert from rectangular \((x, y)\) to polar \((r, \theta)\), use the following formulas:

• \(r = \sqrt{x^2 + y^2}\)
• \(\theta = \arctan\left(\frac{y}{x}\right)\)

This means that the distance from the origin is found by calculating the square root of the sum of the squares of the x and y distances.
The angle is found by taking the arctangent of the ratio of y to x.

Conversely, when converting from polar coordinates \( (r, \theta) \) to rectangular coordinates \((x, y)\), use:

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

These relationships allow for easy conversion between the two coordinate systems, enabling the use of the most convenient system for a given problem.