Problem 44 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to find one set of polar coordinates for the point given in rectangular coordinates. (There are many correct answers.) $$(3 \sqrt{2}, 3 \sqrt{2})$$

Step-by-Step Solution

Answer

The polar coordinates for the point \((3\sqrt{2}, 3\sqrt{2})\) in the rectangular coordinate format are \((6, \frac{\pi}{4})\)

Step 1: Convert rectangular coordinates to polar coordinates

The point \((3 \sqrt{2}, 3 \sqrt{2})\) in rectangular coordinates can be converted to polar coordinates using the formula \(r=\sqrt{x^2+y^2}\) for the radial coordinate, and the formula \(\theta = \arctan\left(\frac{y}{x}\right)\) for the angular coordinate. Remember to adjust the value of \(\theta\) according the quadrant where the point is located. In this case, both \(x\) and \(y\) are positive so the point is in the first quadrant and no adjustment is needed.

Step 2: Calculate r

First, calculate \(r\), which is the length from the origin to the point. \(r = \sqrt{(3\sqrt{2})^2 + (3\sqrt{2})^2} = \sqrt{18+18} = \sqrt{36} = 6\)

Step 3: Calculate theta

Next, calculate \(\theta\), which is the counter-clockwise angle from the positive x-axis to the point. \(\theta = \arctan\left(\frac{3\sqrt{2}}{3\sqrt{2}}\right) = \arctan(1) = \frac{\pi}{4}\) radians

Step 4: Write the result in polar coordinates

The polar coordinates for the point \((3\sqrt{2}, 3\sqrt{2})\) in rectangular coordinate are \((6,\frac{\pi}{4})\)

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, allow us to locate points in a plane using two numbers. These two numbers, usually denoted as \(x\) and \(y\), represent the horizontal and vertical distances from a fixed reference point called the origin, which is at \((0, 0)\). Rectangular coordinates are great for calculations involving perpendicular intersections, and they are commonly used in algebra and calculus.

Key features of rectangular coordinates include:

• Presentation by a pair of values \((x, y)\).
• The origin, where \(x = 0\) and \(y = 0\).
• The x-axis representing horizontal displacement and the y-axis representing vertical displacement.

Rectangular coordinates are very efficient for solving geometric problems on a grid, which is why they are widely featured in educational contexts.

Converting Coordinates

Converting between rectangular coordinates and polar coordinates involves understanding how these two systems relate to each other geometrically. The conversion allows us to express points in different formats, optimizing for particular types of calculations.

Here's how to convert from rectangular to polar coordinates:

• Calculate the distance \(r\) from the origin to the point using the formula \(r = \sqrt{x^2 + y^2}\).
• Determine the angle \(\theta\) with the x-axis using \(\theta = \arctan\left(\frac{y}{x}\right)\). Remember that \(\theta\) needs to be adjusted based on which quadrant the point is in.

These formulas are derived from the relationships in right-angled triangles, using the Pythagorean theorem and trigonometric ratios to transform between coordinate systems efficiently.

Polar Coordinate System

The polar coordinate system represents points using the distance from the origin and an angle from a reference direction, usually the positive x-axis. Instead of \((x, y)\), points are described by \((r, \theta)\), where \(r\) is the radial distance and \(\theta\) is the angular displacement.

Why use polar coordinates?

• They simplify problems involving rotational symmetry.
• They are ideal for complex problems in physics involving circular motion.
• They provide an intuitive framework for dealing with waves and oscillations.

In the context of graphing, polar coordinates offer a perspective that can be more natural for certain types of phenomena, such as those arising in engineering and physical sciences.

Trigonometric Functions

Trigonometric functions are fundamental when dealing with polar coordinates. They relate angles to ratios of sides in right-angled triangles, simplifying the conversion between rectangular and polar systems.

The primary trigonometric functions include:

• Sine (\(\sin\)): relates the opposite side and the hypotenuse in a triangle.
• Cosine (\(\cos\)): relates the adjacent side to the hypotenuse.
• Tangent (\(\tan\)): the ratio of the opposite side to the adjacent side, used directly to find angles in coordinate conversion with \(\arctan\).

These functions help convert angles into x and y displacements and vice versa, supporting efficient transformation of data across different coordinate systems, particularly for solving real-world problems, such as navigation and signal processing.