Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations

Problem 5.3.59

Chapter 5, Problem 5.3.59

In Exercises 59–74, convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation. r = 8

Verified step by step guidance

Recall the relationship between polar and rectangular coordinates: \(x = r \cos\theta\) and \(y = r \sin\theta\), and also \(r^2 = x^2 + y^2\).

Given the polar equation \(r = 8\), square both sides to express in terms of \(r^2\): \(r^2 = 8^2\) which simplifies to \(r^2 = 64\).

Substitute \(r^2\) with \(x^2 + y^2\) to convert the equation into rectangular form: \(x^2 + y^2 = 64\).

Recognize that the rectangular equation \(x^2 + y^2 = 64\) represents a circle centered at the origin with radius 8.

To graph the equation, draw a circle centered at the origin \((0,0)\) with radius 8 units on the rectangular coordinate system.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polar and Rectangular Coordinate Systems

Polar coordinates represent points using a radius and an angle (r, θ), while rectangular coordinates use x and y values. Understanding how these systems describe points in the plane is essential for converting equations between them.

Conversion Formulas Between Polar and Rectangular Coordinates

The key formulas for conversion are x = r cos θ and y = r sin θ, with r = √(x² + y²). These allow translating polar equations into rectangular form by substituting r and θ with expressions involving x and y.

Graphing Rectangular Equations

Once the polar equation is converted, graphing involves plotting the rectangular equation on the Cartesian plane. Recognizing the shape (e.g., circle, line) from the rectangular form helps in accurately sketching the graph.

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In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ.

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In Exercises 53–58, perform the indicated operation(s) and write the result in standard form. (2 + i)² − (3 − i)²

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In Exercises 1–8, add or subtract as indicated and write the result in standard form. 6 − (−5 + 4i) − (−13 − i)

Textbook Question

In Exercises 37–52, perform the indicated operations and write the result in standard form.

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Textbook Question

Convert each rectangular equation to a polar equation that expresses r in terms of θ.

y = 3

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Textbook Question

In Exercises 1–8, parametric equations and a value for the parameter t are given. Find the coordinates of the point on the plane curve described by the parametric equations corresponding to the given value of t. x = 4 + 2 cos t, y = 3 + 5 sin t; t = π/2