Explanation Formula used: Conversion formulae: 1 Given a point with polar coordinates ( r , θ ) , the Cartesian coordinates are as follows. x = r cos θ and y = r sin θ 2 Given a point with Cartesian coordinates P ( x , y ) = P ( r , θ ) , its radial polar coordinates satisfies r 2 = x 2 + y 2 and the coordinate θ is determined using the relation tan θ = y x where the quadrant in which θ lies is determined by the signs of x and y. Calculation: Consider the point P ( − x , y ) which is in second quadrant as shown below in Figure 1. Since the point P ( − x , y ) is in second quadrant, cos θ = − x r and sin θ = y r . So, x = − r cos θ and y = r sin θ . Substitute x = − r cos θ and y = r sin θ in the formula r 2 = x 2 + y 2 as follows. r 2 = ( − r cos θ ) 2 + ( r sin θ ) 2 r 2 = r 2 cos 2 θ + r 2 sin 2 θ r 2 = r 2 ( cos 2 θ + sin 2 θ ) r 2 = r 2 Thus, the formula r 2 = x 2 + y 2 is verified. Substitute x = − r cos θ and y = r sin θ in the formula tan θ = y x as follows. tan ( π 2 + θ ) = r sin θ − r cos θ ( ∵ θ ∈ Q 2 ) − tan θ = − tan θ tan θ = tan θ Thus, the formula tan θ = y x is verified. Therefore, the conversion formulae are verified in the second quadrant. Consider the point P ( − x , − y ) which is in third quadrant as shown below in Figure 2. Since the point P ( − x , − y ) is in third quadrant, cos θ = − x r and sin θ = − y r . So, x = − r cos θ and y = − r sin θ . Substitute x = − r cos θ and y = − r sin θ in the formula r 2 = x 2 + y 2 as follows

Calculus, Single Variable: Early Transcendentals (3rd Edition)

3rd Edition

ISBN: 9780134766850

Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz

Publisher: PEARSON

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Chapter 12.2, Problem 2QC

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To sketch: The different versions of Figure 12.21 with the point P in second, third and fourth quadrants and to verify the conversion formulas in each case.

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Chapter 12.2, Problem 1QC

Chapter 12.2, Problem 3QC

Chapter 12 Solutions

Calculus, Single Variable: Early Transcendentals (3rd Edition)

Ch. 12.1 - Identify the graph generated by the parametric...Ch. 12.1 - Prob. 2QC Ch. 12.1 - Prob. 3QC Ch. 12.1 - Prob. 4QC Ch. 12.1 - Prob. 5QC Ch. 12.1 - Prob. 6QC Ch. 12.1 - Explain how a pair of parametric equations...Ch. 12.1 - Prob. 2E Ch. 12.1 - Prob. 3E Ch. 12.1 - Prob. 4E

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The different versions of Figure 12.21 with the point P in second, third and fourth quadrants and to verify the conversion formulas in each case.

Solution Summary: The author illustrates the different versions of Figure 12.21 with the point P in second, third, and fourth quadrants.

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To sketch: The different versions of Figure 12.21 with the point P in second, third and fourth quadrants and to verify the conversion formulas in each case.

Want to see the full answer?

Chapter 12 Solutions