Explanation Proof: Consider the provided equation x 2 + y 2 = r 2 . From the theorem, if x y â€“coordinate system is rotated θ radians to produce a new x ′ y ′ â€“coordinate system, the rotation equations are, x = x ′ c o s θ − y ′ s i n θ and y = x ' sin θ + y ' cos θ . Multiply x = x ′ c o s θ − y ′ s i n θ by cos θ and y = x ′ sin θ + y ′ cos θ by sin θ to eliminate y ′ term. x c o s θ = x ′ c o s 2 θ − y ′ s i n θ c o s θ and y sin θ = x ′ sin 2 θ + y ′ cos θ sin θ Add both the equations to get, x c o s θ + y sin θ = x ′ c o s 2 θ − y ′ s i n θ c o s θ + x ′ sin 2 θ + y ′ cos θ sin θ = x ′ c o s 2 θ + x ′ sin 2 θ = x ′ c o s 2 θ + sin 2 θ = x ′ Now, multiply x = x ′ c o s θ − y ′ s i n θ by sin θ and y = x ′ sin θ + y ′ cos θ by cos θ to eliminate x ′ term

Calculus Early Transcendentals, Binder Ready Version

11th Edition

ISBN: 9781118883822

Author: Howard Anton, Irl C. Bivens, Stephen Davis

Publisher: WILEY

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Question

System that uses coordinates to uniquely determine the position of points. The most common coordinate system is the Cartesian system, where points are given by distance along a horizontal x-axis and vertical y-axis from the origin. A polar coordinate system locates a point by its direction relative to a reference direction and its distance from a given point. In three dimensions, it leads to cylindrical and spherical coordinates.

Chapter 10.5, Problem 16ES

To determine

To prove: That x′=xcosθ+ysinθ and y′=−xsinθ+ycosθ , where xy â€“coordinate system is rotated θ degrees to produce a new x′y′ â€“coordinate system.

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Chapter 10.5, Problem 15ES

Chapter 10.5, Problem 17ES

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Chapter 10 Solutions

Calculus Early Transcendentals, Binder Ready Version

Ch. 10.1 - Prob. 1QCE Ch. 10.1 - The graph of the curve described by the parametric...Ch. 10.1 - Suppose that a parametric curve C is given by the...Ch. 10.1 - To find dy/dx directly from the parametric...Ch. 10.1 - Let L be the length of the curve x=lnt,y=sint1t An...Ch. 10.1 - (a) By eliminating the parameter, sketch the...Ch. 10.1 - (a) By eliminating the parameter, sketch the...Ch. 10.1 - Sketch the curve by eliminating the parameter, and...Ch. 10.1 - Sketch the curve by eliminating the parameter, and...Ch. 10.1 - Sketch the curve by eliminating the parameter, and...

Knowledge Booster

To prove: That x ′ = x c o s θ + y s i n θ and y ′ = − x sin θ + y cos θ , where x y â€“ coordinate system is rotated θ degrees to produce a new x ′ y ′ â€“coordinate system.

Solution Summary: The author proves that xprime =xcostheta +ysin

To prove: That x′=xcosθ+ysinθ and y′=−xsinθ+ycosθ , where xy â€“coordinate system is rotated θ degrees to produce a new x′y′ â€“coordinate system.

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Chapter 10 Solutions