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Chapter 8: Problem 38

In Exercises 21-40, convert each point given in polar coordinates to exact rectangular coordinates. $$ \left(5,270^{\circ}\right) $$

Short Answer

Expert verified
(0, -5)

Step by step solution

01

Understand Polar Coordinates

Polar coordinates represent a point in the form \((r, \theta)\), where \(r\) is the radial distance from the origin and \(\theta\) is the angle measured in degrees from the positive x-axis.
02

Recognize the Given Coordinates

The problem provides polar coordinates \((5, 270^{\circ})\), where \(r = 5\) is the radial distance and \(\theta = 270^{\circ}\) is the angle from the positive x-axis.
03

Use Conversion Formulas to Find x-coordinate

To convert to rectangular coordinates, use the formula \(x = r \cos(\theta)\). Substitute the given values: \[x = 5 \cos(270^{\circ})\]Since \(\cos(270^{\circ}) = 0\), we find:\[x = 5 \times 0 = 0\]
04

Use Conversion Formulas to Find y-coordinate

Next, use the formula \(y = r \sin(\theta)\). Substitute the given values:\[y = 5 \sin(270^{\circ})\]Since \(\sin(270^{\circ}) = -1\), we have:\[y = 5 \times (-1) = -5\]
05

State the Rectangular Coordinates

Combine the calculated values for \(x\) and \(y\) to express the rectangular coordinates as \((x, y) = (0, -5)\).

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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 method for describing the location of a point on a plane. A point in polar coordinates is given as \((r, \theta)\), where \(r\) and \(\theta\) have distinct roles.- **Radial Distance \(r\):** This is the distance from the origin to the point. It is always a non-negative number, denoting how far the point is from the center of the plane.- **Angle \(\theta\):** This angle is measured in degrees (or radians) and determines the direction from the positive x-axis at which the point lies.When reading polar coordinates like \((5, 270^{\circ})\), the number 5 is how far the point is from the center, and the \(270^{\circ}\) angle tells you the direction you'll be moving from the x-axis. In this context, \(270^{\circ}\) signifies three quarters of the way around the circle, pointing directly downwards along the negative y-axis. Understanding these two components helps in converting these polar points to their rectangular counterparts.
Rectangular Coordinates
Rectangular coordinates, or Cartesian coordinates, denote a point using an \(x\) and \(y\) value, which relate to distances along the horizontal (x-axis) and vertical (y-axis) directions, respectively.- **x-coordinate \(x\):** This specifies the horizontal distance from the origin. It can be positive or negative, showing the direction along the x-axis. Positive values move right, while negative ones move left.- **y-coordinate \(y\):** This specifies the vertical distance from the origin. Positive values move upwards, while negative ones move downwards.For instance, after converting the polar coordinates \((5, 270^{\circ})\) into rectangular coordinates, you get \((x, y) = (0, -5)\). This tells you that the point lies directly on the negative y-axis, 5 units downward from the origin. Rectangular coordinates give a clear way to map a point's precise location in terms of horizontal and vertical placement on a plane.
Trigonometric Conversion Formulas
Converting between polar and rectangular coordinates involves using simple trigonometric relationships. The key formulas used are:- **x-coordinate formula:** \(x = r \cos(\theta)\) - **y-coordinate formula:** \(y = r \sin(\theta)\)These formulas rely on the concepts of cosine and sine from trigonometry:- **Cosine \(\cos(\theta)\):** Relates the adjacent side to the hypotenuse in a right triangle, here giving the horizontal component.- **Sine \(\sin(\theta)\):** Relates the opposite side to the hypotenuse, providing the vertical component.For \((5, 270^{\circ})\), using \(x = 5 \cos(270^{\circ})\), since \(\cos(270^{\circ}) = 0\), we derive that \(x = 0\). Similarly, for \(y = 5 \sin(270^{\circ})\), since \(\sin(270^{\circ}) = -1\), we find \(y = -5\). These formulas effectively translate the radial distance and angular direction into definitive horizontal and vertical components, making it easy to switch between the two systems of coordinates.

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Most popular questions from this chapter

In Exercises 13-40, perform the indicated operation, simplify, and express in standard form. $$ (-3-2 i)(7-4 i) $$

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