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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 5.43
Chapter 5, Problem 5.43

In Exercises 41–43, eliminate the parameter. Write the resulting equation in standard form.


A hyperbola: x = h + a sec t, y = k + b tan t

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1
Start with the given parametric equations of the hyperbola: \(x = h + a \sec t\) and \(y = k + b \tan t\).
Recall the fundamental trigonometric identity relating secant and tangent: \(\sec^2 t - \tan^2 t = 1\).
Express \(\sec t\) and \(\tan t\) in terms of \(x\) and \(y\) by isolating them from the parametric equations: \(\sec t = \frac{x - h}{a}\) and \(\tan t = \frac{y - k}{b}\).
Substitute these expressions into the identity \(\sec^2 t - \tan^2 t = 1\) to eliminate the parameter \(t\).
Simplify the resulting equation to write it in the standard form of a hyperbola: \(\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1\).

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Key Concepts

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

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted as t. Understanding how to manipulate these equations is essential for eliminating the parameter and finding a direct relationship between x and y.
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Trigonometric Identities Involving Secant and Tangent

The identity sec²t - tan²t = 1 is crucial when working with parametric equations involving sec t and tan t. This identity allows the elimination of the parameter t by relating sec t and tan t algebraically.
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Standard Form of a Hyperbola

The standard form of a hyperbola centered at (h, k) is ((x - h)² / a²) - ((y - k)² / b²) = 1. Recognizing this form helps in rewriting the equation after eliminating the parameter to identify the conic section clearly.
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Related Practice
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 = 20(cos 75° + i sin 75°)

z₂ = 4(cos 25° + i sin 25°)

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Textbook Question

In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [4(cos 15° + i sin 15°)]³

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Textbook Question

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

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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/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)

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Textbook Question

In Exercises 53–58, perform the indicated operation(s) and write the result in standard form. (2 − 3i)(1 − i) − (3 − i)(3 + i)

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