Advanced Mathematical Concepts: Precalculus with Applications, Student Edition
Advanced Mathematical Concepts: Precalculus with Applications, Student Edition
1st Edition
ISBN: 9780078682278
Author: McGraw-Hill, Berchie Holliday
Publisher: Glencoe/McGraw-Hill
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Chapter 13.1, Problem 59E
To determine

To calculate: The value of θ for the system of polar equations r=2 and r=2cos2θ . Graph the polar curves and compare the solutions.

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Answer to Problem 59E

The value of θ for the system of polar equations r=2 and r=2cos2θ are (2,180°) and (2,0°) . Graph is provided below,

  Advanced Mathematical Concepts: Precalculus with Applications, Student Edition, Chapter 13.1, Problem 59E , additional homework tip  1

Explanation of Solution

Given information:

The system of polar equations r=2 and r=2cos2θ .

Formula used:

The initial horizontal velocity is vx=vcosθ and initial vertical velocity vy=vsinθ where v is the initial velocity with which object is thrown and θ is the angle it makes with the horizontal.

Calculation:

Consider the system of polar equations r=2 and r=2cos2θ .

Equate both the equations,

  2=2cos2θcos2θ=1cos2θ=cos0°2θ=0°

Simplify it further as,

  θ=0°

Therefore, when θ=0°

  r=2cos2(0°)r=2

Also, one more possible solution is,

  cos2θ=cos360°2θ=360°

Simplify it further as,

  θ=180°

Therefore, when θ=180°

  r=2cos2(180°)r=2

Therefore, two solutions are (2,180°) and (2,0°) .

Sketch the polar curves r=2 and r=2cos2θ .

The curve r=2 is a circle of radius 2 units and the curve r=2cos2θ is a 4 petal rose.

  Advanced Mathematical Concepts: Precalculus with Applications, Student Edition, Chapter 13.1, Problem 59E , additional homework tip  2

The solution of system of polar equations are points of intersection of the two curves.

Thus, the value of θ for the system of polar equations r=2 and r=2cos2θ are (2,180°) and (2,0°) .

Chapter 13 Solutions

Advanced Mathematical Concepts: Precalculus with Applications, Student Edition

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