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Chapter 7: Problem 24

Convert the polar coordinates of each point to rectangular coordinates. $$\left(-5,0^{\circ}\right)$$

Short Answer

Expert verified
Rectangular coordinates: \((-5,0)\).

Step by step solution

01

Understand the given polar coordinates

The given polar coordinates are \((-5, 0^{\circ})\). The first value, -5, is the radius \(r\), and the second value, \(0^{\circ}\), is the angle \(\theta\).
02

Use the conversion formulas

To convert from polar coordinates to rectangular coordinates, use the formulas: \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\).
03

Substitute the values

Substitute \(r = -5\) and \(\theta = 0^{\circ}\) into the formulas: \[x = -5 \cos(0^{\circ}) = -5 \cdot 1 = -5 \] \[y = -5 \sin(0^{\circ}) = -5 \cdot 0 = 0\]
04

Write the rectangular coordinates

The rectangular coordinates are \((-5, 0)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polar Coordinates
Polar coordinates are a way of representing points in a plane using a distance and an angle. The distance is called the radius and is denoted by \(r\), while the angle is measured from the positive x-axis and is denoted by \(\theta\). In the pair \((-5, 0^{\circ})\), -5 is the radius, and \(0^{\circ}\) is the angle. The radius can be positive or negative. A positive radius means the point is in the direction of the angle, while a negative radius means the point is in the opposite direction.
Rectangular Coordinates
Rectangular coordinates, also known as Cartesian coordinates, represent points in a plane using two values: \(x\) and \(y\). These values correspond to the horizontal and vertical distances from the origin (0,0). The pair \((-5, 0)\) means the point is located 5 units to the left of the origin and 0 units up or down. Converting from polar to rectangular coordinates makes it easier to plot or interpret the point in a standard Cartesian plane.
Trigonometric Functions
Trigonometric functions are crucial for converting between polar and rectangular coordinates. The functions \(\text{cosine}\) and \(\text{sine}\) relate an angle in a right triangle to the ratios of two of its sides. To convert polar coordinates to rectangular coordinates, you use:
  • \(x = r \cos(\theta)\)
  • \(y = r \sin(\theta)\)
In this case, substituting \(r = -5\) and \(\theta = 0^{\circ}\) into these formulas:
\[x = -5 \cos(0^{\circ}) = -5 \cdot 1 = -5\]
\[y = -5 \sin(0^{\circ}) = -5 \cdot 0 = 0\]
This gives the rectangular coordinates \((-5, 0)\). The trigonometric functions help translate the point from a circular description to a linear one.

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