Question

In Exercises \(47-52,\) convert the point from spherical coordinates to cylindrical coordinates. $$ (7, \pi / 4,3 \pi / 4) $$

Short answer

The cylindrical coordinates equivalent to the given spherical coordinates are \((7 \cdot \sqrt{2}/2, \pi / 4, -7 \cdot \sqrt{2}/2)\)

Step-by-step solution

Identify the spherical coordinates

The given spherical coordinates are \((7, \pi / 4,3 \pi / 4)\). This means the radial distance from the origin \(r\) is 7, the azimuthal angle \(\theta\) is \(\pi / 4\), and the polar angle \(\phi\) is \(3 \pi / 4\).

Apply the conversion formulas

The conversion formulas from spherical to cylindrical coordinates are: \(r' = r \sin\phi\), \(\theta' = \theta\), and \(z' = r \cos\phi\). Applying these formulas, we get \(r' = 7 \sin(3 \pi / 4)\), \(\theta' = \pi / 4\), and \(z' = 7 \cos(3 \pi / 4)\).

Evaluate the expressions

Evaluating the sine and cosine functions, we obtain: \(r' = 7 \cdot \sqrt{2}/2\), \(\theta' = \pi / 4\), and \(z' = -7 \cdot \sqrt{2}/2\).