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Illustration for Question 38
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39 The square of the sine of an angle plus the square of the cosine of the same angle is always equal to (a) 0 (b) 1 (c) p /2 (d) p (e) 2p 40 Imagine two generic standard-form vectors in xyz space, defined by ordered triples as a = (xa,ya,za) and b = (xb,yb,zb) Now consider the quantity n = xaxb + yayb + zazb What does n represent (a) The dot product of a and b (b) The product of the magnitudes of a and b (c) The sum of the magnitudes of a and b (d) The arithmetic mean of a and b (e) The magnitude of the cross product of a and b 41 Here s a claim concerning coordinate conversions Suppose we have a point (q,r,h ) in cylindrical coordinates We can find the Cartesian x value of this point using the formula x = r cos q The Cartesian y value is y = r sin q The Cartesian z value is z=h What, if anything, is wrong with this claim as stated If anything is wrong with it, how can it be made right (a) The x and y conversions are wrong It should say x = r sin q and y = r cos q (b) The x and y conversions are wrong It should say x = h cos q and y = h sin q
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(c) The z conversion is wrong It should say z = (r 2 + h 2 )1/2 (d) The z conversion is wrong It should say z = h tan q (e) Nothing is wrong with the claim as stated 42 If we graph the unit circle in the Cartesian xy plane and then pick a point (x 0,y0) on that circle such that x0 0, then y0/x0 is equal to (a) the cosine of the counterclockwise angle between the positive x axis and a ray going out from the origin through (x0,y0) (b) the Arccosine of the counterclockwise angle between the positive x axis and a ray going out from the origin through (x0,y0) (c) the tangent of the counterclockwise angle between the positive x axis and a ray going out from the origin through (x0,y0) (d) the Arctangent of the counterclockwise angle between the positive x axis and a ray going out from the origin through (x0,y0) (e) the secant of the counterclockwise angle between the positive x axis and a ray going out from the origin through (x0,y0) 43 Figure FE-7 illustrates a general cylindrical coordinate system Note that the line segment connecting the origin and point P is always perpendicular to the line segment connecting points P and P Based on this knowledge and the information in the diagram, the straight-line distance d between the origin and point P is (a) (r 2 + h 2 )1/2 (b) r sin q (c) r cos q (d) rh cos q (e) impossible to determine unless we have more information
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Illustration for Question 43
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44 The Arctangent function (a) reverses the work of the tangent function (b) is the reciprocal of the tangent function (c) tells us the tangent of an angle measuring an integer multiple of p radians (d) tells us the tangent of an angle measuring an odd-integer multiple of p /2 radians (e) tells us the length of an arc having a measure of a given angle 45 Which of the three variables portray the same geometric dimension in both spherical and cylindrical coordinates (a) The radius (b) The vertical direction angle (c) The radius and the vertical direction angle (d) The horizontal direction angle (e) The horizontal and vertical direction angles 46 In Cartesian coordinates, the point ( 5, 12) is the same distance from the origin as the point (a) (0,17) (b) (10,7) (c) (0, 13) (d) (6,11) (e) All of the above 47 Imagine two generic standard-form vectors in xyz space, defined by ordered triples as a = (xa,ya,za) and b = (xb,yb,zb) Now consider the quantity q = [(xa2 + ya2 + z a2)(x b2 + y b2 + zb2)]1/2 sin qab where qab is the smaller angle between a and b as determined in the plane containing them both What does q represent (a) The dot product of a and b (b) The product of the magnitudes of a and b (c) The ratio of the magnitudes of a and b (d) The Cartesian product of a and b (e) The magnitude of the cross product of a and b
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