Question

In Exercises \(25-28,\) convert the point from spherical coordinates to rectangular coordinates. $$ (5, \pi / 4,3 \pi / 4) $$

Short answer

The converted rectangular coordinates are \((2.5, 2.5, -3.54)\).

Step-by-step solution

Identify the Spherical Coordinates

First, identify the given spherical coordinates. In this case, they are denoted as \( (r, \theta, \phi) = (5, \pi / 4,3 \pi / 4) \).

Apply Spherical to Rectangular Conversion Formula

The general formulas to convert spherical coordinates to rectangular coordinates (denoted as (x, y, z)) are: \[ x= r \sin(\phi) \cos(\theta), \] \[ y= r \sin(\phi) \sin(\theta), \] and \[ z= r\cos(\phi). \] Plug in the given spherical coordinates into each of these formulas.

Calculate x Value

For the x-coordinate, substitute \( r = 5\), \( \phi = 3 \pi / 4 \), and \( \theta = \pi / 4 \) into the x-formula: \[ x= 5 \sin(3\pi / 4) \cos(\pi / 4). \] This simplifies to: \[ x= 5(0.7071)(0.7071) = 2.5. \]

Calculate y Value

For the y-coordinate, again substitute the given values into the y-formula: \[ y= 5 \sin(3\pi / 4) \sin(\pi / 4). \] This simplifies to the same value as x, so \[ y=2.5. \]

Calculate z Value

Finally for z-coordinate, substitute the given values into the z-formula: \[ z= 5\cos(3\pi / 4). \] This simplifies to: \[ z = 5(-0.7071) = -3.54. \]

Final Rectangular Coordinates

Combine the calculated x, y, z values to give the final rectangular coordinates.