Question

Plot the point whose cylindrical coordinates are given. Then find the rectangular coordinates of the point.

a) (2,5π/6,1)

b) (8,2π/3,5

Answer 1

a) Rectangular coordinates:
`\((-√(3), 1, 1)\)`

b) Rectangular coordinates:
`\((-4, 4√(3), 5)\)`

a) To plot the point with cylindrical coordinates (2, 5π/6, 1), first, locate the point on the polar coordinate system. The radial distance is 2, the angular coordinate is 5π/6 (measured counterclockwise from the positive x-axis), and the height (z-coordinate) is 1.

Now, to find the rectangular coordinates (x, y, z), we can use the following conversions:

`\[ x = r \cos(\theta) \]`

`\[ y = r \sin(\theta) \]`

`\[ z = z \]`

Substitute the given values:

`\[ x = 2 \cos\left((5\pi)/(6)\right) \]`

`\[ y = 2 \sin\left((5\pi)/(6)\right) \]`

`\[ z = 1 \]`

Evaluate these expressions to find the rectangular coordinates.

b) For the cylindrical coordinates (8, 2π/3, 5), follow the same process. Plot the point on the polar coordinate system with a radial distance of 8, an angular coordinate of 2π/3, and a height (z-coordinate) of 5. Then, use the conversion formulas to find the rectangular coordinates:

`\[ x = 8 \cos\left((2\pi)/(3)\right) \]`

`\[ y = 8 \sin\left((2\pi)/(3)\right) \]`

`\[ z = 5 \]`

Evaluate these expressions to determine the rectangular coordinates of the point.