Explanation Given: The equation of the conic x 2 + 2 y 2 = 16 in XY-coordinates when the coordinate axes are rotated at the angle ϕ = sin − 1 3 5 . Approach: The x -axes and y-axes in a coordinate plane are rotated through the acute angle ϕ to produce the X -axis and Y -axis. Then, the coordinates ( x , y ) and ( X , Y ) of a point in the x y and the X Y -planes are x = X cos ϕ − Y sin ϕ …… (1) y = X sin ϕ + Y cos ϕ …… (2) Here, ( x , y ) is the coordinates of the point and ϕ is the angle at which coordinate axis are rotated. Write the formula of the inverse trigonometry. sin 2 θ = 1 − cos 2 θ sin ( sin − 1 x ) = x Calculation: Substitute sin − 1 3 5 for ϕ in equation (1). x = X cos ( sin − 1 3 5 ) − Y sin ( sin − 1 3 5 ) = X ( 1 − sin 2 ( sin − 1 3 5 ) ) − Y sin ( sin − 1 3 5 ) = X ( 1 − { sin ( sin − 1 3 5 ) } 2 ) − Y sin ( sin − 1 3 5 ) = X ( 1 − ( 3 5 ) 2 ) − Y ( 3 5 ) Further solve, x = X ( 1 − 9 25 ) − Y ( 3 5 ) = 16 25 X − 3 5 Y = 4 5 X − 3 5 Y Similarly, Substitute sin − 1 3 5 for ϕ in equation (2).

Algebra and Trigonometry (MindTap Course List)

4th Edition

ISBN: 9781305071742

Author: James Stewart, Lothar Redlin, Saleem Watson

Publisher: Cengage Learning

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Chapter 12.5, Problem 12E

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The equation of the conic x2+2y2=16 in XY-coordinates when the coordinate axes are rotated at the angle ϕ=sin−135.

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Chapter 12.5, Problem 11E

Chapter 12.5, Problem 13E

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Chapter 12 Solutions

Algebra and Trigonometry (MindTap Course List)

Ch. 12.1 - A parabola is the set of all points in the plane...Ch. 12.1 - Prob. 2E Ch. 12.1 - Prob. 3E Ch. 12.1 - Prob. 4E Ch. 12.1 - Prob. 5E Ch. 12.1 - Prob. 6E Ch. 12.1 - 5-10 Graphs of Parabolas Match the equation with...Ch. 12.1 - Prob. 8E Ch. 12.1 - 5-10 Graphs of Parabolas Match the equation with...Ch. 12.1 - Prob. 10E

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The equation of the conic x 2 + 2 y 2 = 16 in XY-coordinates when the coordinate axes are rotated at the angle ϕ = sin − 1 3 5 .

Solution Summary: The author calculates the equation of the conic x+2y2=16 in XY-coordinates when the coordinate axes are rotated at the angle.

The equation of the conic x2+2y2=16 in XY-coordinates when the coordinate axes are rotated at the angle ϕ=sin−135.

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Chapter 12 Solutions