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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 71
Chapter 5, Problem 71

In Exercises 59–74, convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation.
r = 6 cos θ + 4 sin θ

Verified step by step guidance
1
Recall the relationships between polar and rectangular coordinates: \(x = r \cos \theta\), \(y = r \sin \theta\), and \(r^2 = x^2 + y^2\).
Start with the given polar equation: \(r = 6 \cos \theta + 4 \sin \theta\).
Multiply both sides of the equation by \(r\) to eliminate the \(r\) on the left side: \(r \cdot r = r (6 \cos \theta + 4 \sin \theta)\), which gives \(r^2 = 6r \cos \theta + 4r \sin \theta\).
Substitute the rectangular coordinate equivalents: \(r^2 = x^2 + y^2\), \(r \cos \theta = x\), and \(r \sin \theta = y\). This transforms the equation into \(x^2 + y^2 = 6x + 4y\).
Rearrange the equation to standard form by bringing all terms to one side: \(x^2 - 6x + y^2 - 4y = 0\). Then complete the square for both \(x\) and \(y\) terms to express the equation in the form of a circle.

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Key Concepts

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

Polar to Rectangular Coordinate Conversion

Polar coordinates (r, θ) relate to rectangular coordinates (x, y) through the formulas x = r cos θ and y = r sin θ. Converting polar equations to rectangular form involves substituting these expressions to rewrite the equation solely in terms of x and y.
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Trigonometric Identities and Relationships

Understanding the relationships between sine, cosine, and their geometric interpretations is essential. For example, recognizing that cos θ = x/r and sin θ = y/r helps in manipulating and simplifying equations during conversion from polar to rectangular form.
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Graphing in the Rectangular Coordinate System

Once the equation is converted to rectangular form, graphing involves plotting points (x, y) on the Cartesian plane. Familiarity with the shapes of conic sections and lines helps interpret the resulting equation and sketch its graph accurately.
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