Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations

Problem 21

Chapter 5, Problem 21

In Exercises 21–40, eliminate the parameter t. Then use the rectangular equation to sketch the plane curve represented by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of t. (If an interval for t is not specified, assume that −∞ < t < ∞. x = t, y = 2t

Verified step by step guidance

Identify the given parametric equations: \(x = t\) and \(y = 2t\).

Express the parameter \(t\) in terms of \(x\) from the first equation: \(t = x\).

Substitute \(t = x\) into the second equation to eliminate the parameter: \(y = 2x\).

Recognize that the rectangular equation \(y = 2x\) represents a straight line with slope 2 passing through the origin.

To sketch the curve, draw the line \(y = 2x\) on the coordinate plane and add arrows pointing in the direction of increasing \(t\) (which corresponds to increasing \(x\) and \(y\) values).

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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, usually t. Instead of y as a direct function of x, both x and y depend on t, allowing representation of more complex curves and motions.

Eliminating the Parameter

Eliminating the parameter involves rewriting the parametric equations to form a single equation in x and y. This is done by solving one equation for t and substituting into the other, converting the parametric form into a rectangular (Cartesian) equation.

Orientation of Parametric Curves

Orientation indicates the direction in which the curve is traced as the parameter t increases. Arrows on the graph show this direction, helping to understand the motion or progression along the curve over the parameter's interval.

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Related Practice

Textbook Question

In Exercises 22–24, find the quotient z₁/z₂ of the complex numbers. Leave answers in polar form.

z₁ = 5 (cos 4π/3 + i sin 4π/3)

z₂ = 10 (cos π/3 + i sin π/3)

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Textbook Question

In Exercises 21–28, divide and express the result in standard form. 2 / 3 - i

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Textbook Question

In Exercises 21–28, divide and express the result in standard form.

3 / 4+i

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Textbook Question

In Exercises 9–20, use point plotting to graph the plane curve described by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of t. x = 2t, y = |t − 1|; −∞ < t < ∞

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Textbook Question

In Exercises 13–34, test for symmetry and then graph each polar equation. r = 1 + 2 cos θ

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Textbook Question

In Exercises 21–26, use a polar coordinate system like the one shown for Exercises 1–10 to plot each point with the given polar coordinates. Then find another representation of this point in which

a. r>0, 2π < θ < 4π.

b. r<0, 0. < θ < 2π.

c. r>0, −2π. < θ < 0.

(5, π/6)

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