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## Spherical to Cylindrical Coordinates Conversion

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Spherical coordinates are used to locate points in three-dimensional (3D) space. They are determined using two angles (theta and phi) and one radial coordinate (r). They are one of two commonly used polar coordinate systems, with the other one being the more widely known cylindrical coordinate system.

The three coordinates of position in spherical coordinates (theta, phi, and r) are defined as follows:

• Theta (θ) – Theta is the angle measured clockwise from the x-axis in the x-y plane. It is measured in degrees.
• Phi (Φ) – Phi is the angle measured counterclockwise from the z-axis. It is measured in degrees.
• Radius (r) – Radius (r) is the radius of the sphere measured from the origin and is measured in the same unit as the x, y and z coordinates.

Using the calculator

• Enter the value of (theta, phi, and r)

• Hit the equal sign to compute the conversion from Spherical to Cylindrical coordinates ### Conversion Made Easy: A Step-By-Step Guide To Converting Spherical Coordinates To Cylindrical Coordinates

Cylindrical coordinates are a three-dimensional coordinate system used to describe the position of a point in a 3D space. They are based on the polar coordinates system and have the same origin. In cylindrical coordinates, each point is represented using a radius, angle, and a height value.

Converting from spherical coordinates to cylindrical coordinates is a straightforward process. In this guide, we’ll breakdown the steps for you.

#### Step 1: Convert the spherical coordinates to rectangular coordinates.

The first step is to convert spherical coordinates (radius, angle, polar angle) to cartesian (rectangular) coordinates. The equations for the conversion are as follows:

x = r * cos(angle) * sin(polar angle)

y = r * sin(angle) * sin(polar angle)

z = r * cos(polar angle)

#### Step 2: Convert the rectangular coordinates to cylindrical coordinates.

Now that we have the rectangular coordinates, we can convert them to cylindrical coordinates. The equations for the conversion are as follows:

Angle = arctan(y/x)

Height = z

#### Step 3: Simplify the equations.

The equations can be simplified in order to make the conversion easier. For example, the equations can be simplified by eliminating the use of the arctan (inverse tangent) function and square roots. The simplified equations are as follows:

Angle = tan-1(y/x)

Height = z

#### Step 4: Check your results.

Once you’ve converted the spherical coordinates to cylindrical coordinates, it’s important to check your results by plotting the points on a graph or using a calculator. This will help you to make sure you’ve done the conversion correctly.

By following the steps outlined in this guide, you’ll be able to easily convert spherical coordinates to cylindrical coordinates in no time. Good luck and enjoy from Mathcrave on YouTube Channel

### A Comprehensive Guide To Working With Spherical Coordinates (And How To Calculate Them)

In order to determine spherical coordinates, one must first convert the Cartesian coordinates (x, y, and z) to spherical coordinates. This can be done in a few simple steps.

Step 1: Find the radius (r). The radius (r) is the length of the position vector from the origin and can be calculated using the following formula:

r = sqrt(x^2 + y^2 + z^2)

Step 2: Find the theta and phi. To find the theta and phi, one must calculate the tangents of theta and phi using the following formulas:

Theta = arctan(y / x)

Phi = arccos(z / r)

Step 3: Convert the theta and phi to degrees. Once the tangents have been found, one must then convert the theta and phi to degrees (in order to express them in spherical coordinates) using the following formula:

Theta (degrees) = Theta (radians) * (180/pi)

Phi (degrees) = Phi (radians) * (180/pi)

Once the three components of the spherical coordinates (theta, phi, and r) have been determined, the point can then be expressed in spherical coordinates.

### Example

Let’s say the point A has Cartesian coordinates (4, 0, 3), and we want to determine point A’s spherical coordinates.

Step 1: Find the radius (r):

r = sqrt(4^2 + 0^2 + 3^2)

r = 5

Step 2: Find the tangents of theta and phi:

Theta = arctan(0 / 4)

Theta = 0

Phi = arccos(3 / 5)

Phi = arccos(0.6)

Step 3: Convert theta and phi to degrees:

Theta (degrees) = 0 * (180/pi)

Theta (degrees) = 0

Phi (degrees) = 0.9272 * (180/pi)

Phi (degrees) = 53.1301 degrees

The spherical coordinates of point A are (0, 53.1301, 5).

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