Questions: Convert the complex number from polar to rectangular form. z=8 cis(pi/3) x + Recall that a complex number in polar form is represented as z=r cis(theta)=r(cos (theta)+i sin (theta)). What are the values of r and theta in the given complex number? How are these values used to determine the rectangular form, z=x+y ?

Solution Steps

To convert the complex number from polar to rectangular form, we need to use the given values of \( r \) and \( \theta \). The polar form of a complex number is given by \( z = r \operatorname{cis}(\theta) = r(\cos(\theta) + i \sin(\theta)) \). Here, \( r = 8 \) and \( \theta = \frac{\pi}{3} \). We can use these values to find the rectangular form \( z = x + yi \) by calculating \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \).

The given complex number in polar form is \( z = 8 \operatorname{cis}\left(\frac{\pi}{3}\right) \). Here, we identify the values:

• \( r = 8 \)
• \( \theta = \frac{\pi}{3} \)

To convert to rectangular form, we use the formulas: \[ x = r \cos(\theta) \quad \text{and} \quad y = r \sin(\theta) \] Substituting the values: \[ x = 8 \cos\left(\frac{\pi}{3}\right) = 8 \cdot \frac{1}{2} = 4 \] \[ y = 8 \sin\left(\frac{\pi}{3}\right) = 8 \cdot \frac{\sqrt{3}}{2} = 4\sqrt{3} \approx 6.9282 \]

The rectangular form of the complex number is given by: \[ z = x + yi = 4 + 6.9282i \]

Final Answer