Problem 20 - College Trigonometry

Question

Convert the point from polar coordinates into rectangular coordinates. $$ (-20,3 \pi) $$

Step-by-Step Solution

Answer

The rectangular coordinates are (20, 0).

Step 1: Understand the Concept

Polar coordinates are given in the form \(r, \theta\), where \(r\) is the radius (distance from the origin) and \(\theta\) is the angle in radians. To convert to rectangular coordinates, we use the formulas: \ x = r \cos(\theta) \ and \ y = r \sin(\theta) \.

Step 2: Apply the Formulas

Given the polar coordinates \((-20, 3\pi)\), we will use \ r = -20 \ and \ \theta = 3\pi \. Substitute these values into the formulas: \( x = -20 \cos(3\pi) \ y = -20 \sin(3\pi) \)

Step 3: Calculate \(x\) Coordinate

Calculate \ x = -20 \cos(3\pi) \. Since \ \cos(3\pi) = -1 \, the calculation becomes:\( x = -20 (-1) = 20 \)

Step 4: Calculate \(y\) Coordinate

Calculate \ y = -20 \sin(3\pi) \. Since \ \sin(3\pi) = 0 \, the calculation becomes:\( y = -20 (0) = 0 \)

Step 5: Formulate the Rectangular Coordinates

Combine the calculated \ x \ and \ y \ values to write the rectangular coordinates: \( (20, 0) \).

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, particularly when converting between different coordinate systems. These functions, namely sine and cosine, allow us to express angles and relate them to points on a coordinate plane. In the context of converting polar coordinates to rectangular coordinates, cosine is used to find the horizontal component, while sine finds the vertical component.

• The cosine function, represented as \( \cos(\theta) \), gives the x-coordinate when you know the angle \( \theta \) and the radius \( r \).
• The sine function, \( \sin(\theta) \), finds the y-coordinate similarly.

These trigonometric ratios become particularly useful as they simplify the process and reduce errors, providing direct results from intuition-based calculations. For instance, in this exercise, knowing \( \cos(3\pi) = -1 \) and \( \sin(3\pi) = 0 \) leads us directly to the rectangular coordinates.

Polar Coordinates

Polar coordinates provide a unique way to locate points on a plane. Unlike the rectangular system that relies on two perpendicular axes, polar coordinates use a central point and an angle.

• Each point is described by the radial distance \( r \) from the origin and the angle \( \theta \) from the positive x-axis.
• In this format, \((-20, 3\pi)\), \( r = -20 \) indicates the point is located 20 units in the opposite direction from the direction implied by the angle.

The negative radius might seem counterintuitive, but it's essentially describing a point directly opposite the positive radial direction at the same angle.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, allow you to pinpoint any location on a two-dimensional plane using two perpendicular axes, usually labeled x (horizontal) and y (vertical).

• To convert from polar to rectangular, the point \((r, \theta)\) is transformed using the formulas: \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \).
• These transformations simplify the description of the point by breaking it into straight-direction movements along the axes.

Upon conversion of \((-20, 3\pi)\) using \( x = -20 \cos(3\pi) = 20 \) and \( y = -20 \sin(3\pi) = 0 \), we obtain the rectangular coordinates \((20, 0)\), illustrating how these conversions seamlessly switch between describing a circle and a grid.