b. Let's figure out how to convert 3D rectangular points into spherical points. This diagram will help. P is some point (r, 8, ). Let's convert it to a Cartesian (i.e., rectangular) point (x, y, z). i. zis the most straightforward. Write an expression for z in terms of 1, 8, and . Hint: start by identifying the line segment on the diagram above that corresponds to the z coordinate of the point. ii. Write an expression for y in terms of r, 8, and . iii. ~ Write an expression for x in terms of r, 0, and . iv. Using your answers, convert the spherical point (4, /3, m/4) into rectangular coordinates. c. How about rectangular to spherical? Let's convert (x, y, z) to a spherical point (t, 0, ). i. ris the most straightforward. Write an expression for 7 in terms of x, y, and z. ii. ~ Write an expression for 0 in terms of x, y, and z. iii. ~ Write an expression for in terms of x, y, and z. iv. Using your answers, convert the Cartesian point (2, 4, 5) into spherical coordinates. d. Let's convert our equations from part a. Convert each equation to Cartesian; explain why you are or are not surprised by the result. i r=6 ii. 06=120 iii. =T1/4 5. What if polar, but 3D? The 3D analogue of polar coordinates is spherical coordinates. A point in space is expressed as (r, 0, @) where r and 6 mean the same thing they did in polar (distance from the pole and angle from the positive x axis respectively) and is an angle relative to the z axis. 4 x(n 0, ) X (NB: spherical coordinate notation is not always consistent. The \"math\" convention is to keep the meaning of 6 the same in 2D as in 3D, and that's what I'1l use... if you're looking at a physics/engineering textbook, though, you will probably see that the meanings of 8 and are reversed so, the angle relative to the x axis would be instead of 0. You will also likely find sources that use p to refer to what used to be the distance between the pole and the point in question.) a. What type of graph is produced by... i r=6? . 0=120? ii. @=m/4? Explain each answer. You may find it helpful to explore these graphs in Geogebra or something similar