Question

In Exercises \(1-4,\) convert the point from cylindrical coordinates to rectangular coordinates. $$ (4,7 \pi / 6,3) $$

Short answer

The rectangular coordinates for the given cylindrical coordinates are: \((-2 \sqrt{3}, -2, 3)\)

Step-by-step solution

Identify the cylindrical coordinates

The given cylindrical coordinates are \((r, θ, z) = (4, 7π/6, 3)\). The given values for \(r\), \(θ\), and \(z\) are 4, \(\frac{7 \pi}{6}\), and 3 respectively.

Conversion of r and θ to x and y

The conversion of r and θ to x and y is done using the formulas for conversion from cylindrical to rectangular coordinates, which are \(x=r \cos \theta\) and \(y=r \sin \theta\). Substituting the given values, we get: \(x=4 \cos \frac{7π}{6} = -2 \sqrt{3}\) and \(y=4 \sin \frac{7π}{6} = -2\).

Applying the coordinate for z

In cylindrical coordinates, the z-coordinate remains the same. Hence, \(z = z = 3\).