Textbook Question
In Exercises 45–52, find the quotient z₁/z₂ of the complex numbers. Leave answers in polar form. In Exercises 49–50, express the argument as an angle between 0° and 360°.
z₁ = cos 80° + i sin 80°
z₂ = cos 200° + i sin 200°
Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Blitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblem 5.3.58
Blitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblem 5.3.58Chapter 5, Problem 5.3.58
In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ.
x² = 6y
Verified step by step guidance1
Recall the relationships between rectangular coordinates (x, y) and polar coordinates (r, \(\theta\)):
\(x = r \cos{\theta}\)
\(y = r \sin{\theta}\)
Substitute \(x = r \cos{\theta}\) and \(y = r \sin{\theta}\) into the given equation \(x^2 = 6y\) to rewrite it in terms of \(r\) and \(\theta\):
\((r \cos{\theta})^2 = 6 (r \sin{\theta})\)
Simplify the equation:
\(r^2 \cos^2{\theta} = 6r \sin{\theta}\)
Since \(r\) appears on both sides, and \(r = 0\) is a trivial solution, divide both sides by \(r\) (assuming \(r \neq 0\)) to isolate \(r\):
\(r \cos^2{\theta} = 6 \sin{\theta}\)
Finally, solve for \(r\) to express it explicitly in terms of \(\theta\):
\(r = \frac{6 \sin{\theta}}{\cos^2{\theta}}\)
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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 (x, y) represent points on a plane using horizontal and vertical distances, while polar coordinates (r, θ) represent points by their distance from the origin and 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 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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06:17Convert Points from Polar to Rectangular
Expressing r in Terms of θ
After substituting x and y with r cos θ and r sin θ, the goal is to manipulate the equation algebraically to isolate r as a function of θ. This often involves factoring and simplifying to achieve a clear polar equation.
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6:36Simplifying Trig Expressions
Related Practice
Textbook Question
In Exercises 53–56, find two different sets of parametric equations for each rectangular equation. y = 4x − 3
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Textbook Question
In Exercises 29–36, simplify and write the result in standard form.
√3² − 4 ⋅ 2 ⋅ 5
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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Textbook Question
In Exercises 21–28, divide and express the result in standard form.
2+3i / 2+i
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Textbook Question
In Exercises 45–52, use your answers from Exercises 41–44 and the parametric equations given in Exercises 41–44 to find a set of parametric equations for the conic section or the line.
Ellipse: Center: (−2,3); Vertices: 5 units to the left and right of the center; Endpoints of Minor Axis: 2 units above and below the center
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