Skip to main content
Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblem 12.2.11
Chapter 12, Problem 12.2.11

9–13. Graph the points with the following polar coordinates. Give two alternative representations of the points in polar coordinates.


(-1, -π/3)

Verified step by step guidance
1
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.
Given the point \((-1, -\frac{\pi}{3})\), note that the radius \(r\) is negative. A negative radius means the point is located in the direction opposite to the angle \(\theta\).
To find an alternative representation, convert the point by changing the radius to positive and adjusting the angle by adding \(\pi\) (180 degrees) to the original angle: \((r, \theta) = (-1, -\frac{\pi}{3})\) becomes \((1, -\frac{\pi}{3} + \pi)\).
Another alternative is to add \(2\pi\) to the angle to keep the radius negative but express the angle in a positive coterminal form: \((-1, -\frac{\pi}{3} + 2\pi)\).
Finally, you can also express the point with a positive radius and an angle coterminal to the one found in step 3 by adding or subtracting multiples of \(2\pi\) to the angle, for example \((1, -\frac{\pi}{3} + \pi + 2\pi)\).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1m
Was this helpful?

Key Concepts

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

Polar Coordinates System

The polar coordinate system represents points in a plane using a radius and an angle, denoted as (r, θ). The radius r is the distance from the origin, and θ is the angle measured from the positive x-axis. Understanding how to plot points using these two values is fundamental for graphing in polar coordinates.
Recommended video:
05:32
Intro to Polar Coordinates

Negative Radius in Polar Coordinates

A negative radius means the point is plotted in the direction opposite to the angle θ. For example, (−r, θ) is equivalent to (r, θ + π), reflecting the point across the origin. Recognizing this helps in converting and graphing points with negative radii correctly.
Recommended video:
05:32
Intro to Polar Coordinates

Alternative Representations of Polar Coordinates

Polar points can have multiple equivalent representations by adding or subtracting 2π to the angle or changing the sign of the radius while adjusting the angle by π. This flexibility allows expressing the same point in different forms, which is useful for understanding and comparing polar coordinates.
Recommended video:
05:32
Intro to Polar Coordinates
Related Practice
Textbook Question

Second derivative Assume a curve is given by the parametric equations x=f(t) and y=g(t), where f and g are twice differentiable. Use the Chain Rule to show that y″x=(fʹ(t)g″(t)−gʹ(t)f″(t))/(fʹ(t))³.  

Textbook Question

57–64. Graphing polar curves Graph the following equations. Use a graphing utility to check your work and produce a final graph.


r² = 4 sin θ  

Textbook Question

Air drop—inverse problem A plane traveling horizontally at 100 m/s over flat ground at an elevation of 4000 m must drop an emergency packet on a target on the ground. The trajectory of the packet is given by

x = 100t, y = −4.9t² + 4000, t ≥ 0

where the origin is the point on the ground directly beneath the plane at the moment of the release. How many horizontal meters before the target should the packet be released in order to hit the target?

1
views
Textbook Question

63–66. Tracing hyperbolas and parabolas Graph the following equations. Then use arrows and labeled points to indicate how the curve is generated as θ increases from 0 to 2π. 


r = 3/(1 - cos θ)

1
views
Textbook Question

45–60. Areas of regions Find the area of the following regions.


The region inside the rose r = 4 sin 2θ and inside the circle r = 2

Textbook Question

37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.


The left half of the parabola y=x ² +1, originating at (0, 1)

1
views