Convert the point \( (\rho, \theta, \phi)=\left(2, \frac{7 \pi}{4}, \frac{5 \pi}{6}\right) \) to Cartesian coordinates. Give answers either as expressions, or decimals to at least one decimal place. \( (x, y, z)=\square \)

To convert from spherical coordinates \((\rho, \theta, \phi)\) to Cartesian coordinates \((x, y, z)\), we use the following formulas: \[ x = \rho \sin(\phi) \cos(\theta) \] \[ y = \rho \sin(\phi) \sin(\theta) \] \[ z = \rho \cos(\phi) \] For the given point \((\rho, \theta, \phi)=\left(2, \frac{7 \pi}{4}, \frac{5 \pi}{6}\right)\), we can substitute these values: 1. Calculating \(z\): \[ z = 2 \cos\left(\frac{5\pi}{6}\right) = 2 \cdot \left(-\frac{\sqrt{3}}{2}\right) = -\sqrt{3} \approx -1.732 \] 2. Calculating \(x\): \[ x = 2 \sin\left(\frac{5\pi}{6}\right) \cos\left(\frac{7\pi}{4}\right) = 2 \cdot \left(\frac{1}{2}\right) \cdot \left(-\frac{\sqrt{2}}{2}\right) = -\frac{\sqrt{2}}{2} \approx -0.707 \] 3. Calculating \(y\): \[ y = 2 \sin\left(\frac{5\pi}{6}\right) \sin\left(\frac{7\pi}{4}\right) = 2 \cdot \left(\frac{1}{2}\right) \cdot \left(-\frac{\sqrt{2}}{2}\right) = -\frac{\sqrt{2}}{2} \approx -0.707 \] Thus, the Cartesian coordinates are: \[ (x, y, z) \approx \left(-0.707, -0.707, -1.732\right) \] So the final answer is \( (x, y, z)=\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}, -\sqrt{3}\right) \) or approximately \((x, y, z) \approx (-0.707, -0.707, -1.732)\).