Problem 59 - Precalculus

Question

A point in rectangular coordinates is given. Convert the point to polar coordinates. $$(5,12)$$

Step-by-Step Solution

Answer

The polar coordinates of the point given in rectangular system as (5,12) are approximately \((13, 1.176) \) or \((13, 67.38^{\circ})\).

Step 1: Identify the Rectangular Coordinates

The given rectangular coordinates are (5,12). In the rectangular system, these are represented as (x, y).

Step 2: Calculate the Radius (r)

In polar coordinates, we represent a point as (r, θ), where r is the distance of the point from the origin. This distance or radius r is given by the formula \(r = \sqrt{x^2 + y^2}\). In our case, this becomes \(r = \sqrt{5^2 + 12^2} \).\nLet's compute this.

Step 3: Calculate the Angle (θ)

For the angle θ, we use the formula \( θ = \arctan(\frac{y}{x}) \). In our case, this becomes \( θ = \arctan(\frac{12}{5}) \). Let's compute this ensuring that we are in the correct quadrant.

Step 4: Converting the Angle to Proper Form

Make sure that the angle is between 0 and \(2π\). If not, add or subtract \(2π\) until it is.

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, involve a pair of numbers that specify the location of a point on a plane. Each point is defined by an ordered pair, usually written as \((x, y)\).
The first number, \(x\), represents the horizontal distance from the origin, and the second number, \(y\), represents the vertical distance. These coordinates help us find the exact position of a point within a 2D space using a straightforward grid-like system.
Unlike polar coordinates, which use an angle and a distance, rectangular coordinates rely on these two perpendicular axes (usually labeled \(x\) and \(y\)). This system is extremely useful for solving many types of mathematical problems, especially those involving geometry and algebra.

Radius Calculation

In polar coordinates, the radius \(r\) signifies the distance of a point from the origin (0,0) of the coordinate system. The formula to compute \(r\) is given by:\[r = \sqrt{x^2 + y^2}\]This formula is derived from the Pythagorean theorem, where \(x\) and \(y\) are the horizontal and vertical distances from the origin respectively.
For example, if you have a point located at \((5,12)\) in rectangular coordinates, the radius \(r\) would be calculated as follows:\[r = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13\]This result tells us that the point is 13 units away from the origin.

Angle Calculation

The angle \(\theta\) in polar coordinates indicates the direction of the point relative to the positive \(x\)-axis. To calculate \(\theta\), the formula used is:\[\theta = \arctan\left(\frac{y}{x}\right)\]This formula gives us the angle in radians, which is the standard unit of angular measurement in mathematics. Since angles can overflow past 360 degrees or \(2\pi\) radians, they can often be adjusted to be within a customary range (such as from 0 to \(2\pi\) radians).
In the provided example of \((5,12)\), the angle would be calculated as:\[\theta = \arctan\left(\frac{12}{5}\right) \]Make sure the calculated angle is in the proper range, ensuring the angle represents the correct direction on the plane.

Coordinate Conversion

Coordinate conversion is the process of changing from one coordinate system to another, such as from rectangular to polar coordinates, or vice versa. This conversion is incredibly useful in fields like engineering, physics, and computer graphics, where different systems can offer various advantages depending on the problem at hand.
To convert from rectangular \((x, y)\) to polar coordinates \((r, \theta)\), we use the following steps:

• Calculate the radius \(r\) using \(r = \sqrt{x^2 + y^2}\).
• Determine the angle \(\theta\) with \(\theta = \arctan\left(\frac{y}{x}\right)\).
• Ensure \(\theta\) falls within the desired range, often 0 to \(2\pi\).

By transforming the point \((5,12)\), the polar coordinates become \((13, \theta)\), where \(\theta\) is calculated explicitly using the angle formula. Knowing both coordinate systems allows for flexibility and precision in mathematical and practical applications.