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Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblema 71
Capitolo 5, Problema 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

Guida verificata passo dopo passo
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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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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Convert Points from Polar to Rectangular

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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Fundamental Trigonometric Identities

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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Introduction to Graphs & the Coordinate System
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