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Spherical Coordinates

Spherical coordinates of the system is denoted as (ρ, θ, Φ). This coordinate system mainly used in three dimensional systems. In three dimensional systems spherical coordinate system are used for finding the surface area. These are also called as spherical polar coordinates. ρ is the radius of the system. θ is inclination angle and Φ is azimuth angle.

Rectangular coordinates to spherical coordinates:

ρ = √ (x2 + y2 + z2)

θ = cos-1(x / ρ)

Φ = cos-1 (x / (ρ * sin θ))

Spherical coordinates to rectangular coordinates:

x = ρ (sin θ) (cos Φ)

y = ρ (sin θ) (sin Φ)

z = ρ (cos θ)

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Converting Spherical to Rectangular Coordinates

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Here is an example Problem of converting spherical coordinates to rectangular coordinates provided to understand better:

Problem:

Convert the spherical coordinates (32, 68°, 74°) into rectangular coordinates

Solution:

Given spherical coordinates are, ρ = 32, θ = 68°, Φ = 74°

Convert the above values into rectangular coordinates using the formula,

x = ρ (sin θ) (cos Φ)

y = ρ (sin θ) (sin Φ)

z = ρ (cos θ)

Substitute the above values in the given formulas, we get

x = 32 * (sin 68°) (cos 74°)

x = 8.17

y = 32 * (sin 68°) (sin 74°)

y = 28.51

z = 32 cos 68°

z = 11.98

Answer:

The rectangular coordinates are x = 8.17, y = 28.51, z = 11.98

Converting Rectangular to Spherical Coordinates

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Below is an example showing how to convert rectangular coordinates to spherical coordinates:

Example 1:

Convert the rectangular coordinates (21, 28, 32) into spherical coordinates.

Solution:

Given x = 21, y = 28, z = 32

Formula:

ρ = √ (x2 + y2 + z2)

θ = cos-1(x / ρ)

Φ = cos-1 (x / (ρ * sin θ))

Substitute the given x, y, z values in the given formula, we get

ρ = √ (212 + 282 + 322)

ρ = 47.42

θ = cos-1(x / ρ)

= cos-1(21 / 47.42)

= 63.71°

Φ = cos-1 (x / (ρ * sin θ)

= cos-1 (21 / (47.42 * sin 63.71°))

= 60.42°

Answer:

The spherical coordinates are 47.42, 63.71°, 60.42°

Spherical Coordinates Practice Problems

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Example 1:

Convert the spherical coordinates (12, 45°, 60°) into rectangular coordinates.

Answer:

The rectangular coordinates are x = 4.24, y = 7.34, z = 8.48

Example 2:

Convert the spherical coordinates (6, 30°, 65°) into rectangular coordinates.

Answer:

The rectangular coordinates are x =1.26, y = 2.7, z = 5.19

Example 3:

Convert the rectangular coordinates (7, 12, 4) into spherical coordinates.

Answer:

The spherical coordinates are (14.45, 61.02°, 56.37 )

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