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When we input the values for x and y to the formula for r, we get r = [( 4)2 + 12]1/2 = (16 + 1)1/2 = 171/2 Because z = 0, we know that h=z=0 The exact cylindrical equivalent point is (q,r,h) = {[p + Arctan ( 1/4)],171/2,0} Using a calculator set for radians to approximate the angle coordinate to four decimal places, we get q = p + Arctan ( 1/4) 31416 + ( 02450) 28966 When we approximate the radius coordinate to four decimal places, we obtain r = 171/2 41231 The approximate ordered triple representing our point in cylindrical coordinates is therefore (q,r,h) (28966,41231,0) The first coordinate represents an angle in radians The second and third coordinates represent linear displacements in space 5 We want to find the (x,y,z) equivalent of (q,r,h) = (p /4,21/2,1) First, let s find x Using the cylindrical-to-Cartesian conversion equation, we get x = r cos q = 21/2 cos (p /4) = 21/2 21/2/2 = 2/2 = 1 The cylindrical-to-Cartesian conversion equation for y tells us that y = r sin q = 21/2 sin (p /4) = 21/2 21/2/2 = 2/2 = 1 Finding z involves no conversion at all We have simply z=h=1 Therefore, the Cartesian equivalent point of the cylindrical (q,r,h) = (p /4,21/2,1) is (x,y,z) = (1,1,1) That s what we began with when we worked out the example in the chapter text
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510 Worked-Out Solutions to Exercises: 1-9
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6 In spherical coordinates, the graph of the equation q = 0 appears as a vertical plane containing the reference axis In xyz space, this would be the xz plane It s exactly the same situation as we had in the cylindrical coordinate system when we solved Problem 1, because the horizontal direction angles are identical in both systems The graph of f = 0 in spherical coordinates is a vertical straight line that coincides with the Cartesian z axis The graph of r = 0 in spherical coordinates is the origin point In xyz space, it s (0,0,0) 7 Figure A-9 is a plot of the point (q,f,r) = (3p /4,p /4,8) in spherical coordinates 8 We have a point in spherical three-space whose coordinates are given by P = (q,f,r) = (p /4,0,1) The formula for x is x = r sin f cos q When we plug in the spherical values, we get x = 1 sin 0 cos (p /4) = 1 0 21/2/2 = 0 The formula for y is y = r sin f sin q Plugging in the spherical values, we get y = 1 sin 0 sin (p /4) = 1 0 21/2/2 = 0
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Illustration for the solution to Problem 7 in Chap 9 Each radial division represents 1 unit
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The formula for z is z = r cos f Plugging in the spherical values, we get z = 1 cos 0 = 1 1 = 1 Therefore, the coordinates in xyz space are P = (0,0,1) 9 We want to convert the xyz space point ( 4,1,0) to spherical coordinates To find the radius, we use the formula r = (x2 + y2 + z2)1/2 Plugging in the values, we get r = [( 4)2 + 12 + 02]1/2 = (16 + 1 + 0)1/2 = 171/2 To find the horizontal angle, we use the formula q = p + Arctan ( y /x) because x < 0 and y > 0 When we plug in the values for x and y, we get q = p + Arctan [1/( 4)] = p + Arctan ( 1/4) To find the vertical angle, we can use the formula f = Arccos (z /r) We already know that r = 171/2, so f = Arccos (0/171/2) = Arccos 0 = p /2 Our spherical ordered triple, listing the coordinates in the order P = (q,f,r), is P = {[p + Arctan ( 1/4)],p /2,171/2} Using a calculator set for radians and rounding the irrational values to four decimal places, we get q = p + Arctan ( 1/4) 31416 + ( 02450) 28966 f = p /2 31416 / 2 15708 r = 171/2 41231
512 Worked-Out Solutions to Exercises: 1-9
Therefore, we can approximate the spherical coordinates as P (28966,15708,41231} 10 We re given the point P in cylindrical three-space as P = (q,r,h) = [3p /4,61/2/2,61/2/2] Our first task is to find the equivalent coordinates in xyz space Here are the conversion formulas once again, for reference: x = r cos q y = r sin q z=h Plugging in the numbers to these formulas gives us x = 61/2/2 cos (3p /4) = 61/2/2 ( 21/2/2) = 31/2/2 y = 61/2/2 sin (3p /4) = 61/2/2 21/2/2 = 31/2/2 z = h = 61/2/2 Therefore, we have the Cartesian equivalent point P = (x,y,z) = ( 31/2/2,31/2/2,61/2/2) When we check this against the intermediate result we got as we solved the last challenge in the chapter text, we see that the two agree So far, we re doing okay! Now let s convert this Cartesian ordered triple to spherical coordinates To find the spherical radius, we use the formula r = (x2 + y2 + z2)1/2 Plugging in the values, we get r = [( 31/2/2)2 + (31/2/2)2 + (61/2/2)2]1/2 = (3/4 + 3/4 + 6/4)1/2 = (12/4)1/2 = 31/2 To find the horizontal angle, we use the formula q = p + Arctan ( y /x) because x < 0 and y > 0 When we plug in the values for x and y, we get q = p + Arctan [31/2/( 31/2)] = p + Arctan ( 1) = p + ( p /4) = 3p /4 To find the vertical angle, we can use the formula f = Arccos (z /r)
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