Skip to main content
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 5.3.54
Chapter 5, Problem 5.3.54

In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ.


x² + y² = 16

Verified step by step guidance
1
Recall the relationship between rectangular coordinates (x, y) and polar coordinates (r, \(\theta\)): \(x = r \cos\theta\) and \(y = r \sin\theta\).
Substitute \(x = r \cos\theta\) and \(y = r \sin\theta\) into the given equation \(x^{2} + y^{2} = 16\).
This substitution gives the equation \((r \cos\theta)^{2} + (r \sin\theta)^{2} = 16\).
Simplify the equation by factoring out \(r^{2}\): \(r^{2}(\cos^{2}\theta + \sin^{2}\theta) = 16\).
Use the Pythagorean identity \(\cos^{2}\theta + \sin^{2}\theta = 1\) to simplify to \(r^{2} = 16\), then solve for \(r\) to express it in terms of \(\theta\).

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.

Rectangular and Polar Coordinate Systems

Rectangular coordinates represent points using (x, y) on a plane, while polar coordinates use (r, θ), where r is the distance from the origin and θ is the angle from the positive x-axis. Understanding how these systems relate is essential for converting equations between them.
Recommended video:
05:32
Intro to Polar Coordinates

Relationship Between Rectangular and Polar Coordinates

The key formulas connecting the two systems are x = r cos θ and y = r sin θ. Additionally, r² = x² + y². These relationships allow substitution of x and y in terms of r and θ to rewrite rectangular equations in polar form.
Recommended video:
06:17
Convert Points from Polar to Rectangular

Converting Equations to Express r in Terms of θ

To convert a rectangular equation to polar form expressing r as a function of θ, substitute x and y with r cos θ and r sin θ, respectively, then solve algebraically for r. This process isolates r, showing how the radius varies with the angle θ.
Recommended video:
6:50
Convert Equations from Polar to Rectangular
Related Practice
Textbook Question

In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ. x² + y² = 9

Textbook Question

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

( −6 − √(−12)) / 48

5
views
Textbook Question

In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [2(cos 10° + i sin 10°)]³

2
views
Textbook Question

Evaluate x² − 2x + 2 for x = 1 + i.

Textbook Question

In Exercises 53–58, perform the indicated operation(s) and write the result in standard form. (2 − 3i)(1 − i) − (3 − i)(3 + i)

6
views
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.


Line: Passes through (−2,4) and (1,7)

1
views