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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 5.3.52
Chapter 5, Problem 5.3.52

Convert each rectangular equation to a polar equation that expresses r in terms of θ.
y = 3

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1
Recall the relationship between rectangular coordinates \((x, y)\) and polar coordinates \((r, \theta)\): \(x = r \cos\theta\) and \(y = r \sin\theta\).
Given the rectangular equation \(y = 3\), substitute \(y\) with its polar form: \(r \sin\theta = 3\).
To express \(r\) in terms of \(\theta\), isolate \(r\) by dividing both sides of the equation by \(\sin\theta\): \(r = \frac{3}{\sin\theta}\).
Note that this expression is valid for values of \(\theta\) where \(\sin\theta \neq 0\), since division by zero is undefined.
Thus, the polar equation expressing \(r\) in terms of \(\theta\) corresponding to the line \(y = 3\) is \(r = \frac{3}{\sin\theta}\).

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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 the relationship between these systems 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). These allow expressing rectangular variables x and y in terms of polar variables r and θ, enabling the transformation of equations from rectangular to polar form.
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Convert Points from Polar to Rectangular

Expressing r in Terms of θ

To convert an equation like y = 3 into polar form, substitute y with r sin(θ) and solve for r. This process isolates r as a function of θ, which is the goal when expressing polar equations explicitly.
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Simplifying Trig Expressions
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