3. Consider the Cartesian equationx² - y^= 2x²y - 2xy²,show that its polar form is given by the equationT=sin(20)cos(0) + sin(0)

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Answered: 3. Consider the Cartesian equation x -…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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1. Match each equation in rectangular coordinates with its equation in polar coordinates: (a) x² + y² = 4 O r²(1-2 sin² 0) = 4 r(cos 0 + sin() = 4 r = 2 sin 0 Or=2 (b) x² + (y − 1)² = 1 O r²(1-2 sin²0) = 4 r(cos+ sin 0) = 4 r=2 sin 0 r = 2 (c) x+y=4 Or² (1-2 sin² 0) = 4 r(cos+ sin 0) = 4 r = 2 sin 0 Or=2
2. Write +i tan A in polar/trinometric form r(cos 0+ i sin 0) or re" or r Z0.
4. Match each equation in rectangular coordinates with its equation in polar coordinates. (a) x² + y² = 4 (b) x²+(y-1)=1 (c) x² - y² = 4 (d) x+y=4 (i) r²(1-2 sin²0) = 4 == (ii) r(cos + sin 0) = 4 (iii) r = 2 sin (iv) r=2
As the wheel of radius r cm in the figure rotates, the rod of length L attached to point P drives a piston back and forth in a straight line. Let x be the distance from the origin to point at the end of the rod as shown. (a) Use the Pythagorean Theorem to show that L² = (x − r cos 0)² + ² sin² 0. (b) Differentiate the equation in part (a) with respect to t to show that 0=2(x-r cos 0) (d+rsin 0df)+2r² sin cos de. dt (c) Calculate the speed of the piston when , assuming that r = 10 cm, L = 30 cm, and the wheel rotates at 4 revolutions per minute. L X = Piston moves back and forth e
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3 Find z' in polar form. z = 4[cos(25°) + i · sin(25°)] 23 cis( %3|
Find the rectangular equation of the polar curve defined by r = sin20. A. x +y-2xy=0 B. (x + y}-2xy = 0 C. x* +y-x = 0 D. x* +y* - y = 0 E. x*+y-xy = 0 O A O B O D O E

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3. Consider the Cartesian equation x - y = 2x²y - 2xy², show that its polar form is given by the equation sin (20) T= cos(0)+sin(0)