Problem 5 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 3-8: Change the polar coordinates to rectangular coordinates. (a) \((8,-2 \pi / 3)\) (b) \((-3,5 \pi / 3)\)

Step-by-Step Solution

Answer

(a) Rectangular: \\((-4, -4\sqrt{3})\\); (b) Rectangular: \\((-\frac{3}{2}, \frac{3\sqrt{3}}{2})\\)

Step 1: Understand the Polar Coordinates

Polar coordinates are given in the form \(r, \theta\), where \(r\) is the radial distance from the origin, and \(\theta\) is the angle measured from the positive x-axis. In this exercise, we need to convert the given polar coordinates into rectangular coordinates \( (x, y) \).

Step 2: Recall the Conversion Formulas

To convert from polar to rectangular coordinates, use the formulas: \(\begin{align*}x &= r \cos(\theta) \y &= r \sin(\theta)\end{align*}\) These will allow us to find the rectangular coordinates \( (x, y) \).

Step 3: Convert (a) \\( (8, -\frac{2\pi}{3}) \\)

First, calculate \(x\): \(x = 8 \cos(-\frac{2\pi}{3}) = 8 \cos(-120^\circ)= 8 \times -\frac{1}{2} = -4\)Next, calculate \(y\): \(y = 8 \sin(-\frac{2\pi}{3}) = 8 \sin(-120^\circ) = 8 \times -\frac{\sqrt{3}}{2} = -4\sqrt{3}\)Therefore, the rectangular coordinates are \((-4, -4\sqrt{3})\).

Step 4: Convert (b) \\((-3, \frac{5\pi}{3})\\)

First, calculate \(x\): \(x = -3 \cos(\frac{5\pi}{3}) = -3 \cos(300^\circ) = -3 \times \frac{1}{2} = -\frac{3}{2}\)Next, calculate \(y\): \(y = -3 \sin(\frac{5\pi}{3}) = -3 \sin(300^\circ) = -3 \times -\frac{\sqrt{3}}{2} = \frac{3\sqrt{3}}{2}\)Therefore, the rectangular coordinates are \((-\frac{3}{2}, \frac{3\sqrt{3}}{2})\).

Key Concepts

Polar Coordinates

Polar coordinates are a system used to specify the locations of points in a plane using a pair called \( (r, \theta) \). Here, \( r \) represents the radial distance from a fixed point known as the origin, and \( \theta \) is the angle formed with the positive direction of the x-axis. This system is particularly useful in contexts where the relationships between the points are better expressed using angles and distances rather than x and y values.
In many physics or engineering problems, polar coordinates help to simplify complex shape descriptions. Imagine the address of a place where instead of using road names and numbers, we're using directions (angle) and distance (radial) from a common landmark.

• Polar coordinates are written as \( (r, \theta) \).
• \( r \) must be a non-negative value, though negative values can indicate direction reversal.
• The angle \( \theta \) is typically measured in radians, where a full circle is \( 2\pi \) radians or 360 degrees.

Mastering polar coordinates enhances your ability to solve problems in various fields, enabling you to express and understand concepts by angles and distances effectively.

Rectangular Coordinates

Rectangular coordinates, often called Cartesian coordinates, describe points using coordinates (x, y). This familiar system was named after Rene Descartes, who laid the foundation for merging geometry and algebra. Cartesian coordinates allow you to find positions on an x-y grid, visually representing horizontal and vertical distances from the origin, \((0, 0)\).
Rectangular coordinates are vital for graphing lines, curves, and shapes in a straightforward manner, where each movement along the grid is an axis-aligned shift.

• The x-coordinate defines the horizontal distance from the origin.
• The y-coordinate specifies the vertical distance from the origin.
• Together, \((x, y)\) tells you the precise location of any point on a plane.

Understanding rectangular coordinates is essential for basically any aspect of geometry or algebra, making it a fundamental tool in mathematical problem-solving.

Trigonometric Functions

Trigonometric functions, commonly known as trig functions, play a crucial role in coordinate conversion. They relate the angles of a triangle to the lengths of its sides and are foundational for converting between polar and rectangular coordinates.
Key trigonometric functions used in this context are sine (\(\sin\)) and cosine (\(\cos\)).
To convert from polar to rectangular coordinates, you use these formulas:

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

Each equation takes advantage of the unit circle's principles, effectively translating radial and angular directions into horizontal and vertical measures.
Trigonometry as it is used here simplifies the process of moving between different coordinate systems, which is a critical skill in both academic and real-world applications.