Question

Convert from rectangular to spherical coordinates. (Use symbolic notation and fractions where needed. Give your answer as a point's coordinates in the form (*,*,*)

(-5√2,5√2, 10√3) = ________

Answer 1

To convert from rectangular to spherical coordinates, use the formulas x = r sin(θ) cos(φ), y = r sin(θ) sin(φ), and z = r cos(θ). Given the rectangular coordinates (-5√2, 5√2, 10√3), substitute the values into the formulas to obtain the spherical coordinates (20, π/6, −π/4).

Step-by-step explanation:

To convert from rectangular to spherical coordinates, we can use the formulas:
x = r sin(θ) cos(φ)
y = r sin(θ) sin(φ)
z = r cos(θ)

Given the rectangular coordinates (-5√2, 5√2, 10√3), we can substitute the values into the formulas to obtain the spherical coordinates.

r = √(x² + y² + z²) = √((-5√2)² + (5√2)² + (10√3)²) = √(50 + 50 + 300) = √400 = 20

θ = arccos(z/r) = arccos(10√3/20) = arccos(√3/2) = π/6

φ = arctan(y/x) = arctan((5√2)/(−5√2)) = arctan(-1) = −π/4

Therefore, the spherical coordinates for the given rectangular coordinates are (20, π/6, −π/4).

Answer 2

The rectangular coordinates
`\((-5√(2), 5√(2), 10√(3))\)`can be expressed in spherical coordinates.

Step-by-step explanation:

To convert rectangular coordinates
`\((-5√(2), 5√(2), 10√(3))\)`to spherical coordinates
`\((r, \theta, \phi)\),` we use the formulas:

`\[ r = √(x^2 + y^2 + z^2) \]`

`\[ \theta = \tan^(-1)\left((y)/(x)\right) \]`

`\[ \phi = \cos^(-1)\left((z)/(r)\right) \]`

For this specific point, the values are:

`\[ r = \sqrt{(-5√(2))^2 + (5√(2))^2 + (10√(3))^2} = √(100 + 100 + 300) = 10 \]`

`\[ \theta = \tan^(-1)\left((5√(2))/(-5√(2))\right) = (3\pi)/(4) \]`

`\[ \phi = \cos^(-1)\left((10√(3))/(10)\right) = \cos^(-1)(√(3)) = (\pi)/(6) \]`

Therefore, the spherical coordinates are \((10, \frac{3\pi}{4}, \frac{\pi}{6})\), and in the form requested, it is \((10, \frac{3\pi}{4}, \tan^{-1}\left(\frac{5}{-5}\right))\).