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 19

Chapter 5, Problem 19

In Exercises 11–20, use a polar coordinate system like the one shown for Exercises 1–10 to plot each point with the given polar coordinates. (−2, − π/2)

Verified step by step guidance

Recall that a point in polar coordinates is given as \((r, \theta)\), where \(r\) is the radius (distance from the origin) and \(\theta\) is the angle measured from the positive x-axis (polar axis).

Note that the given point is \((-2, -\frac{\pi}{2})\). The negative radius means we first consider the point at radius \(2\) but in the opposite direction of the angle \(-\frac{\pi}{2}\).

Since the angle \(-\frac{\pi}{2}\) corresponds to rotating \(\frac{\pi}{2}\) radians clockwise from the positive x-axis, identify this direction on the polar coordinate system (which points downward along the negative y-axis).

Because the radius is negative, move in the opposite direction of the angle \(-\frac{\pi}{2}\), which means moving \(2\) units in the direction of \(-\frac{\pi}{2} + \pi = \frac{\pi}{2}\) (upward along the positive y-axis).

Plot the point \(2\) units away from the origin along the positive y-axis, which corresponds to the adjusted angle \(\frac{\pi}{2}\), completing the plotting of the point with coordinates \((-2, -\frac{\pi}{2})\).

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Key Concepts

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

Polar Coordinates

Polar coordinates represent points in a plane using a distance from the origin (radius r) and an angle θ measured from the positive x-axis. Each point is expressed as (r, θ), where r can be positive or negative, and θ is typically in radians or degrees.

Negative Radius in Polar Coordinates

A negative radius means the point is plotted in the direction opposite to the angle θ. For (−r, θ), you move |r| units along the line at angle θ + π (180 degrees), effectively reflecting the point across the origin.

Angle Measurement and Direction

Angles in polar coordinates are measured counterclockwise from the positive x-axis. Negative angles indicate clockwise rotation. For example, an angle of −π/2 corresponds to a 90-degree rotation clockwise from the positive x-axis.

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Related Practice

Textbook Question

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

Textbook Question

In Exercises 11–26, plot each complex number. Then write the complex number in polar form. You may express the argument in degrees or radians. −3

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Textbook Question

In Exercises 19–21, find the product of the complex numbers. Leave answers in polar form.

z₁ = 3(cos 40°+i sin 40°)

z₂ = 5(cos 70°+i sin 70°)

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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 < ∞

Textbook Question

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

Textbook Question

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