To determine the quadrant in which the complex number [tex]\(8\left(\cos \frac{2 \pi}{3} + i \ln \frac{2 \pi}{3}\right)\)[/tex] lies, we need to analyze the real and imaginary parts of the complex number step-by-step.
### Step 1: Calculate [tex]\(\cos \frac{2\pi}{3}\)[/tex]
[tex]\[
\cos \frac{2\pi}{3} = -0.5
\][/tex]
### Step 2: Calculate [tex]\(\ln \frac{2\pi}{3}\)[/tex]
[tex]\[
\ln \frac{2\pi}{3} \approx 0.7393
\][/tex]
### Step 3: Express the complex number
The given complex number can be written in the form:
[tex]\[
8 \left(\cos \frac{2\pi}{3} + i \ln \frac{2\pi}{3}\right)
\][/tex]
### Step 4: Separate into real and imaginary parts
Multiply both parts by 8:
[tex]\[
8 \left(-0.5 + i \cdot 0.7393\right) = 8 \cdot -0.5 + 8i \cdot 0.7393
\][/tex]
[tex]\[
= -4 + i \cdot 5.9144
\][/tex]
So, the complex number is:
[tex]\[
-4 + 5.9144i
\][/tex]
### Step 5: Determine the quadrant
- The real part is [tex]\(-4\)[/tex]. Since it is negative, it means the complex number is on the left side of the imaginary axis.
- The imaginary part is [tex]\(5.9144\)[/tex]. Since it is positive, it means the complex number is above the real axis.
Given that the complex number has a negative real part and a positive imaginary part, it lies in the second quadrant (II).
### Conclusion
The complex number [tex]\(8\left(\cos \frac{2 \pi}{3} + i \ln \frac{2 \pi}{3}\right)\)[/tex] lies in the second quadrant.
