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Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook
All textbooksBlitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblem 5.3.56
Chapter 5, Problem 5.3.56

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


x² + (y + 3)² = 9

Verified step by step guidance
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Recall the relationships between rectangular coordinates (x, y) and polar coordinates (r, \(\theta\)): \(x = r \cos\theta\) and \(y = r \sin\theta\).
Substitute \(x = r \cos\theta\) and \(y = r \sin\theta\) into the given equation \(x^2 + (y + 3)^2 = 9\) to rewrite it in terms of \(r\) and \(\theta\).
Expand the expression: \(x^2\) becomes \((r \cos\theta)^2 = r^2 \cos^2\theta\), and \((y + 3)^2\) becomes \((r \sin\theta + 3)^2\).
Write the equation as \(r^2 \cos^2\theta + (r \sin\theta + 3)^2 = 9\) and expand the squared term: \((r \sin\theta + 3)^2 = r^2 \sin^2\theta + 6r \sin\theta + 9\).
Combine like terms and simplify the equation to isolate \(r\) in terms of \(\theta\). Use the Pythagorean identity \(\cos^2\theta + \sin^2\theta = 1\) to simplify \(r^2\) terms.

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Key Concepts

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

Rectangular and Polar Coordinate Systems

Rectangular coordinates represent points using (x, y) on a plane, while polar coordinates use (r, θ), where r is the distance from the origin and θ is the angle from the positive x-axis. Understanding how these systems relate is essential for converting equations between them.
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Intro to Polar Coordinates

Conversion Formulas Between Rectangular and Polar Coordinates

The key formulas are x = r cos θ and y = r sin θ, which allow substitution of rectangular variables with polar expressions. These formulas enable rewriting equations from rectangular form into polar form by expressing x and y in terms of r and θ.
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Convert Points from Polar to Rectangular

Equation of a Circle in Polar Coordinates

A circle centered at (h, k) with radius a in rectangular form is (x - h)² + (y - k)² = a². When converting to polar, substitute x and y with r cos θ and r sin θ, then simplify to express r as a function of θ, which may involve algebraic manipulation to isolate r.
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Intro to Polar Coordinates
Related Practice
Textbook Question

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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Textbook Question

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

Textbook Question

In Exercises 1–10, indicate if the point with the given polar coordinates is represented by A, B, C, or D on the graph. (3, π)

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Textbook Question

In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. (√3 − i)⁶

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Textbook Question

In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [1/2 (cos π/10 + i sin π/10)]⁵

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