Problem 82 - Algebra and Trigonometry

Question

Find the rectangular coordinates of each pair of points. Then find the distance, in simplified radical form,between the points. $$ (6, \pi) \text { and }\left(5, \frac{7 \pi}{4}\right) $$

Step-by-Step Solution

Answer

The rectangular coordinates of the points are (-6, 0) and \((- \frac{5\sqrt{2}}{2}, -\frac{5\sqrt{2}}{2})\). The distance between these points in simplest radical form is \(5\sqrt{2}\).

Step 1: Conversion to Rectangular Coordinates

For a pair of points given in polar coordinates, the conversion to rectangular coordinates is made through the formulas \(x=r \cos \theta\) and \(y=r \sin \theta\). Therefore, the rectangular coordinates of these two points: (6, π) and (5, 7π/4) will be (-6, 0) and \((- \frac{5\sqrt{2}}{2}, - \frac{5\sqrt{2}}{2})\)

Step 2: Application of Distance Formula

The distance formula is \(d= \sqrt{ (x2 - x1)² + (y2 - y1)² }\). Therefore, the distance will be: \(\sqrt{ \left( - \frac{5\sqrt{2}}{2} - (-6) \right)^2 + \left(- \frac{5\sqrt{2}}{2} - 0 \right)^2 }\).

Step 3: Final Calculation and Simplification of the Result

After applying the distance formula, simplify the resulting expression to obtain the simplest form. Upon simplification, the distance is hence of the form \(5\sqrt{2}\).

Key Concepts

Distance Formula in Coordinates

Understanding the distance formula in coordinates is crucial for measuring the straight-line distance between two points in a Cartesian plane. The formula is derived from the Pythagorean theorem in a right triangle and is given by:
\[ d = \sqrt{ (x_2 - x_1)^2 + (y_2 - y_1)^2 } \]
Where \(d\) represents the distance, \((x_1, y_1)\) are the coordinates of the first point, and \((x_2, y_2)\) are the coordinates of the second point. To calculate the distance between two points in a rectangular (Cartesian) coordinate system, you subtract the coordinates of the first point from the corresponding coordinates of the second point, square the differences, add the squared differences together, and take the square root of that sum. This formula is a vital tool in various fields of mathematics, including geometry, trigonometry, and calculus, as well as in real-world applications like GPS navigation and robotics.

Converting Polar to Cartesian Coordinates

Polar and Cartesian coordinates are two different systems for describing locations on a plane. To convert polar coordinates, which are given as \((r, \theta)\), to Cartesian coordinates \((x, y)\), you can use the following trigonometric formulas:
\[ x = r \cos(\theta) \]\[ y = r \sin(\theta) \]
In polar coordinates, \(r\) represents the distance from the origin to the point, and \(\theta\) is the angle formed with the positive x-axis. The conversion is important because while polar coordinates are better for circular and spiral patterns, many mathematical functions and operations require the Cartesian system. By converting, we can perform these operations and later convert back if needed. This skill is especially useful in fields like engineering and physics, where switching between coordinate systems can simplify problem-solving.

Simplifying Radical Expressions

Simplifying radical expressions is a process to express the square root (or other roots) in its simplest form. The main goal is to find the largest square factor of the number inside the radical and remove it outside the radical. This is because the square root of a square number is always a whole number. Here's an example of simplifying a squared radical:
If you have \(\sqrt{50}\), you can simplify it as \(\sqrt{25 \times 2}\), which then becomes \(5\sqrt{2}\) because \(\sqrt{25}\) is 5. Simplification becomes essential when dealing with exact values in mathematics, especially in geometry and algebra, as it makes the numbers more manageable and the expressions easier to work with. Sophisticated calculations often require the simplification of radicals to ensure accuracy and ease of understanding.