Question

Find z1z2 if z1 = 3(cos37° + isin37°) and z2 = 2/3(cos53° + isin53°).

Answer 1

Answer:

$z_{1}\cdot z_{2} = 2\cdot (\cos 90^{\circ} + i \cdot \sin 90^{\circ})$

Step-by-step explanation:

Both variable can be rewritten into polar form:

$z_{1} = 3\cdot e^{i\cdot 0.205\pi}$ and $z_{2} = \frac{2}{3}\cdot e^{i\cdot 0.294\pi}$

The complex product is equal to:

$z_{1}\cdot z_{2} = (3)\cdot \left(\frac{2}{3}) \cdot e^{i\cdot (0.205\pi+0.294\pi)}$

$z_{1}\cdot z_{2} = 2 \cdot e^{i\cdot 0.499\pi}}$

The resultant expression in rectangular form is:

$z_{1}\cdot z_{2} = 2\cdot (\cos 90^{\circ} + i \cdot \sin 90^{\circ})$

Answer 2

Answer:

The correct question is:

Find $z_{1}.z_{2}$ if $z_{1}= 3(cos37 + isin37)$ and $z_{2}= \frac{2}{3} (cos53 + isin53)$.

(Note: the angles mentioned in the equations above are in degrees)

The answer is $z_{1}.z_{2}= 2i$

Step-by-step explanation:

$z_{1}.z_{2}= 3(cos37 + isin37)* \frac{2}{3} (cos53 + isin53)$

$z_{1}.z_{2}= (3cos37 + 3isin37)* (\frac{2}{3} cos53 + \frac{2}{3}isin53)$

$z_{1}.z_{2}= 3cos37*\frac{2}{3} cos53 + 3cos37*\frac{2}{3}isin53+3isin37*\frac{2}{3} cos53+3isin37*\frac{2}{3}isin53$

$z_{1}.z_{2}= 0.961+1.275i+0.724i-0.961$ (Because $i*i=i^{2}=-1$ & $i=\sqrt{-1}$)

$z_{1}.z_{2}= 0.961+1.275i+0.724i-0.961$

$z_{1}.z_{2}= 2i$