Table of contents

0. Review of College Algebra4h 43m
1. Measuring Angles40m
2. Trigonometric Functions on Right Triangles2h 5m
3. Unit Circle1h 19m
4. Graphing Trigonometric Functions1h 19m
5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
6. Trigonometric Identities and More Equations2h 34m
7. Non-Right Triangles1h 38m
8. Vectors2h 25m
9. Polar Equations2h 5m
10. Parametric Equations1h 6m
11. Graphing Complex Numbers1h 7m

9. Polar Equations

Polar Coordinate System

Trigonometry

9. Polar Equations

Polar Coordinate System

Previous problemNext problem

1:28 minutes

Problem 19

Textbook Question

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

Understand that polar coordinates are given in the form \((r, \theta)\), where \(r\) is the radial distance from the origin and \(\theta\) is the angle measured from the positive x-axis.

Identify the given polar coordinates: \((-2, -\frac{\pi}{2})\). Here, \(r = -2\) and \(\theta = -\frac{\pi}{2}\).

Since \(r\) is negative, the point is located in the opposite direction of the angle \(\theta\).

Convert the angle \(-\frac{\pi}{2}\) to its equivalent positive angle by adding \(2\pi\) if necessary, or recognize that \(-\frac{\pi}{2}\) corresponds to \(270^\circ\) in standard position.

Plot the point by moving 2 units in the direction opposite to \(-\frac{\pi}{2}\), which is equivalent to moving 2 units in the direction of \(\frac{\pi}{2}\) or \(90^\circ\).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

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 two-dimensional space using a distance from a reference point (the origin) and an angle from a reference direction (usually the positive x-axis). The format is (r, θ), where 'r' is the radial distance and 'θ' is the angle in radians. Understanding this system is crucial for plotting points accurately in polar graphs.

Negative Radius in Polar Coordinates

In polar coordinates, a negative radius indicates that the point is located in the opposite direction of the angle specified. For example, the point (−2, −π/2) means to move 2 units in the direction opposite to the angle of −π/2 (which points downward), effectively placing the point at (2, π/2) in the Cartesian coordinate system.

Conversion Between Polar and Cartesian Coordinates

To plot points in a Cartesian coordinate system, one can convert polar coordinates (r, θ) to Cartesian coordinates (x, y) using the formulas x = r * cos(θ) and y = r * sin(θ). This conversion is essential for visualizing polar points on a standard x-y graph, allowing for a better understanding of their positions relative to the axes.