Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 13 - Vector Geometry - 13.7 Cylindrical and Spherical Coordinates - Exercises - Page 700: 40

Answer

$(2,\frac{\pi}{3},\frac{\pi}{3})$.

Work Step by Step

Since $ x=\frac{\sqrt3}{2}, \, y=\frac{3}{2}, \, z=1$ and $$\rho=\sqrt{x^2+y^2+z^2},\quad \theta=\tan^{-1}( y/x), \quad \phi=\cos^{-1}(z/\rho),$$ then we have $$\rho=\sqrt{\frac{3}{4}+\frac{9}{4}+1^2}= 2, $$ $$\theta=\tan^{-1}( 3/\sqrt 3)=\frac{\pi}{3}, $$ $$\phi=\cos^{-1}(1/2)=\frac{\pi}{3}.$$ Hence, the spherical coordinates are $(2,\frac{\pi}{3},\frac{\pi}{3})$.
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