Certainly! Let's tackle the given problem step-by-step.
We have two complex numbers in polar form:
[tex]\[ z_1 = 7\left(\cos 30^\circ + i \sin 30^\circ\right) \][/tex]
[tex]\[ z_2 = 8\left(\cos 60^\circ + i \sin 60^\circ\right) \][/tex]
We want to find the product of these two complex numbers:
[tex]\[ z_1 \times z_2 \][/tex]
When multiplying two complex numbers in polar form, we multiply their magnitudes and add their angles.
### Step 1: Calculate the Magnitude
The magnitude of the product [tex]\( r \)[/tex] is given by:
[tex]\[ r = r_1 \times r_2 \][/tex]
[tex]\[ r = 7 \times 8 \][/tex]
[tex]\[ r = 56 \][/tex]
### Step 2: Calculate the Angle
The angle of the product [tex]\( \theta \)[/tex] is given by:
[tex]\[ \theta = \theta_1 + \theta_2 \][/tex]
[tex]\[ \theta = 30^\circ + 60^\circ \][/tex]
[tex]\[ \theta = 90^\circ \][/tex]
### Step 3: Write the Product in Polar Form
The product in polar form is:
[tex]\[ 56 \left( \cos 90^\circ + i \sin 90^\circ \right) \][/tex]
### Step 4: Expand the Expression
[tex]\[ 56 \left( \cos 90^\circ + i \sin 90^\circ \right) = 56 (\cos 90^\circ + i \sin 90^\circ) \][/tex]
From trigonometric values:
[tex]\[ \cos 90^\circ = 0 \][/tex]
[tex]\[ \sin 90^\circ = 1 \][/tex]
Thus:
[tex]\[ 56 (\cos 90^\circ + i \sin 90^\circ) = 56 (0 + i \cdot 1) = 56i \][/tex]
So, the real part is [tex]\( 0 \)[/tex] and the imaginary part is [tex]\( 56 \)[/tex].
### Step 5: Summary
Finally, we can summarize our results in the required format:
[tex]\[ 56 \left( \cos 90^\circ + i \sin 90^\circ \right) = 0 + 56i \][/tex]
Therefore, we fill the blanks as follows:
[tex]\[
\begin{aligned}
&{\left[7\left(\cos 30^\circ + i \sin 30^\circ\right)\right]\left[8\left(\cos 60^\circ + i \sin 60^\circ\right)\right] } \\
&= 56 \left(\cos 90^\circ + i \sin 90^\circ\right) \\
&= 0 + 56i
\end{aligned}
\][/tex]
So the completed form is:
[tex]\[
\begin{aligned}
{\left[7\left(\cos 30^\circ + i \sin 30^\circ\right)\right]\left[8\left(\cos 60^\circ + i \sin 60^\circ\right)\right] } & = \square\left(\cos \square^\circ + i \sin \square^\circ\right) \\
& = \square^\circ + \square i
\end{aligned}
\][/tex]
[tex]\[
\begin{aligned}
{\left[7\left(\cos 30^\circ + i \sin 30^\circ\right)\right]\left[8\left(\cos 60^\circ + i \sin 60^\circ\right)\right] } & = 56\left(\cos 90^\circ + i \sin 90^\circ\right) \\
& = 0 + 56i
\end{aligned}
\][/tex]
