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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 5.3.49
Chapter 5, Problem 5.3.49

In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ. 3x + y = 7

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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 rectangular equation \(3x + y = 7\) to get \(3(r \cos{\theta}) + r \sin{\theta} = 7\).
Factor out \(r\) from the left side: \(r(3 \cos{\theta} + \sin{\theta}) = 7\).
Solve for \(r\) by dividing both sides by \((3 \cos{\theta} + \sin{\theta})\): \(r = \frac{7}{3 \cos{\theta} + \sin{\theta}}\).
This expression gives the polar form of the equation, expressing \(r\) explicitly in terms of \(\theta\).

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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 how these systems relate is essential for converting equations between them.
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Intro to Polar Coordinates

Conversion Formulas Between Coordinates

The key formulas for conversion are x = r cos(θ) and y = r sin(θ). These allow substitution of rectangular variables with polar expressions, enabling the transformation of equations from rectangular to polar form.
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Intro to Polar Coordinates

Expressing r in Terms of θ

After substituting x and y with r cos(θ) and r sin(θ), the goal is to isolate r on one side of the equation. This often involves algebraic manipulation to express r explicitly as a function of θ, which is the standard form for polar equations.
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Simplifying Trig Expressions
Related Practice
Textbook Question

In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ. (x − 2)² + y² = 4

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

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In Exercises 65–68, find all the complex roots. Write roots in polar form with θ in degrees. The complex square roots of 9(cos 30° + i sin 30°)

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In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. (1 + i)⁵

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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/12 + i sin π/12)]⁶

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Textbook Question

In Exercises 53–58, perform the indicated operation(s) and write the result in standard form. ___ ___ 5√(−16) + 3√(−81)