Question

In Exercises \(47-52,\) convert the point from spherical coordinates to cylindrical coordinates. $$ (8,7 \pi / 6, \pi / 6) $$

Short answer

The cylindrical coordinates equivalent of the given spherical coordinates (8, 7π/6, π/6) are approximately (4, 7π/6, 6.93).

Step-by-step solution

Convert θ

The value of θ (the azimuthal angle) is the same in both spherical and cylindrical coordinates. So, θ can be directly adapted into its cylindrical counterpart. The given value is 7π/6.

Convert ρ and φ to r and z

The radial distance r in the xy-plane and the height z can be obtained by using these spherical to cylindrical coordinates conversion formulas: \r = ρ*sin(φ) \and z = ρ*cos(φ). Given values for ρ and φ are 8 and π/6, respectively. By substituting, r = 8*sin(π/6) = 4, and z = 8*cos(π/6) = approx. 6.93.

Key concepts

Spherical Coordinates

Spherical coordinates are a system for representing points in three-dimensional space using three numbers: the radial distance (ρ), the polar angle (φ), and the azimuthal angle (θ). Here's how they work:

• The radial distance, ρ, measures how far the point is from the origin, much like the radius in a circle but extended out into 3D space.
• The polar angle, φ, is the angle between the positive z-axis and the line formed from the origin to the point. It's like the latitude on a globe.
• The azimuthal angle, θ, is similar to longitude, and it represents the angle in the xy-plane from the positive x-axis.

In the scenario provided, the point is defined by the coordinates (8, 7π/6, π/6). The number 8 represents the radial distance ρ, 7π/6 is the azimuthal angle (θ), and π/6 is the polar angle (φ). Understanding these components is crucial for converting to cylindrical coordinates.

Cylindrical Coordinates

In contrast, cylindrical coordinates describe a point in space with a different trio of values: radial distance (r), azimuthal angle (θ), and height (z). Similar to polar coordinates in two dimensions, cylindrical coordinates are extended into the third dimension:

• The radial distance (r) indicates the distance from the point to the z-axis, measured in the xy-plane.
• The azimuthal angle (θ) here is the same as in spherical coordinates and represents the angle from the positive x-axis within the xy-plane.
• The height (z) measures how far the point is above or below the xy-plane, analogous to an elevation or altitude.

Transitioning from spherical coordinates to cylindrical is a matter of reinterpreting these spatial measurements, where the radial and polar angles (ρ, φ) get converted to new values for r and z.

Coordinate Conversion

Converting between spherical and cylindrical coordinates requires the use of trigonometric functions to relate the different measurements. The process involves these key equations:

• The azimuthal angle (θ) remains the same in both systems.
• The cylindrical radial distance (r) is computed by multiplying the spherical radius (ρ) with the sine of the polar angle (φ), expressed as r = ρ*sin(φ).
• The cylindrical height (z) is calculated by multiplying the spherical radius (ρ) with the cosine of the polar angle (φ), denoted as z = ρ*cos(φ).

Using the given example, where ρ = 8, θ = 7π/6, and φ = π/6, we can apply the formulas for r and z to get r = 8*sin(π/6) = 4 and z = 8*cos(π/6), which is approximately 6.93. Such conversions are fundamental in fields that require changing perspectives or dimensions, like engineering and physics.

Trigonometric Functions

Trigonometric functions are mathematical functions based on angles, commonly used in the conversion between coordinate systems. They are sine (sin), cosine (cos), and tangent (tan), among others. Here's a brief overview of the functions relevant to our conversion:

• Sine (sin): In a right triangle, it represents the ratio of the length of the side opposite the angle to the length of the hypotenuse.
• Cosine (cos): It indicates the ratio of the length of the adjacent side to the angle to the hypotenuse.

These functions are essential when converting spherical coordinates to cylindrical coordinates. In the exercise given, the sin and cos functions help transition the polar angle φ to the new z and r values in the cylindrical system. The sine of π/6 is 0.5, and the cosine is approximately 0.866, contributing to the calculations of r and z.