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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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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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Fundamental Trigonometric Identities

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, 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.


Hyperbola: Vertices: (4,0) and (−4,0); Foci: (6,0) and (−6,0)

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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°)

Textbook Question

In Exercises 37–52, perform the indicated operations and write the result in standard form.

(3√(−5) )( −4√(−12) )

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Textbook Question

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

y = 3

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Textbook Question

In Exercises 1–8, parametric equations and a value for the parameter t are given. Find the coordinates of the point on the plane curve described by the parametric equations corresponding to the given value of t. x = 4 + 2 cos t, y = 3 + 5 sin t; t = π/2

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.


Circle: Center: (3,5); Radius: 6

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