Problem 29 - Algebra and Trigonometry

Question

Find the rectangular coordinates for the point whose polar coordinates are given. $$ (\sqrt{2},-\pi / 4) $$

Step-by-Step Solution

Answer

The rectangular coordinates are (1, -1).

Step 1: Understand Polar to Rectangular Conversion

Polar coordinates \((r, \theta)\) consist of a radius \(r\) and an angle \(\theta\). They are converted to rectangular coordinates \((x, y)\) using these formulas:\[x = r \cos \theta\] \[y = r \sin \theta\] Here, \(r = \sqrt{2}\) and \(\theta = -\pi/4\).

Step 2: Calculate the x-coordinate

Substitute \(r = \sqrt{2}\) and \(\theta = -\pi/4\) into the formula for the x-coordinate: \[x = \sqrt{2} \cos(-\pi/4)\].Calculate \(\cos(-\pi/4)\), which is \(\sqrt{2}/2\). Then, \[x = \sqrt{2} \times \sqrt{2}/2 = 2/2 = 1\].

Step 3: Calculate the y-coordinate

Substitute \(r = \sqrt{2}\) and \(\theta = -\pi/4\) into the formula for the y-coordinate: \[y = \sqrt{2} \sin(-\pi/4)\].Calculate \(\sin(-\pi/4)\), which is \(-\sqrt{2}/2\). Then, \[y = \sqrt{2} \times (-\sqrt{2}/2) = -2/2 = -1\].

Step 4: Write the Rectangular Coordinates

Combine the calculated x and y values to form the rectangular coordinates: \((x, y) = (1, -1)\).

Key Concepts

Rectangular Coordinates

Rectangular coordinates are a way to describe the position of a point in the Cartesian plane. In this system, any point is defined by a pair of values:

• the x-coordinate (horizontal distance from the origin)
• the y-coordinate (vertical distance from the origin)

These coordinates make it easier to plot points and visualize mathematical problems, such as functions and geometric shapes.
The rectangular or Cartesian coordinate system is widely used in mathematics and engineering because it translates real-world scenarios into numbers easily understandable by computers and humans alike.
A point represented as (x, y) in rectangular coordinates can tell you exactly where the point lies on the plane based on these axis displacements.

Polar Coordinates

Polar coordinates provide a different way to represent points in the plane. Instead of using horizontal and vertical distances, polar coordinates describe points based on:

• the distance from a reference point (the origin) known as the radius, denoted as \( r \)
• the angle from a reference direction (usually the positive x-axis), denoted as \( \theta \)

These coordinates are particularly useful in scenarios involving rotation or when dealing with curves such as spirals or circles.
For example, moving a point around a circle only requires changing the angle \( \theta \), while keeping \( r \) constant. This system is handy in fields like physics and engineering when examining phenomena like waves or circular motion.

Trigonometric Functions

Trigonometric functions play an essential role in converting between polar and rectangular coordinates. They include sine, cosine, and tangent functions, which are based on right-angled triangles.

• The cosine function \( \cos \theta \) relates the angle \( \theta \) to the ratio of the length of the adjacent side to the hypotenuse in a right triangle.
• The sine function \( \sin \theta \) connects \( \theta \) to the ratio of the opposite side's length to the hypotenuse.

These functions help in determining the x and y components from the radius and angle in polar coordinates. In essence, they decompose a vector into its horizontal and vertical parts.
When you use these trigonometric identities for conversion, the radius and angle determine the exact location of the point in rectangular form, resulting in a seamless transformation between the two coordinate systems.