To solve the problem and get the real and imaginary parts of the given expression, we need to analyze it step by step. Let’s break it down in detail:
The expression given is:
[tex]\[ e^{i \pi / 3} = (1 + i) \cdot \left(\cos \frac{\pi}{3} + i \sin \frac{\pi}{3}\right) \][/tex]
First, we need to find the values of [tex]\(\cos \frac{\pi}{3}\)[/tex] and [tex]\(\sin \frac{\pi}{3}\)[/tex]:
- [tex]\(\cos \frac{\pi}{3} = \frac{1}{2}\)[/tex]
- [tex]\(\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}\)[/tex]
Now, we substitute these values into the expression:
[tex]\[ (1 + i) \cdot \left(\frac{1}{2} + i \cdot \frac{\sqrt{3}}{2}\right) \][/tex]
Next, expand the product by distributing [tex]\( (1 + i) \)[/tex]:
[tex]\[ (1 + i) \cdot \left(\frac{1}{2} + i \cdot \frac{\sqrt{3}}{2}\right) = (1 \cdot \frac{1}{2} + 1 \cdot i \cdot \frac{\sqrt{3}}{2}) + (i \cdot \frac{1}{2} + i^2 \cdot \frac{\sqrt{3}}{2}) \][/tex]
Since [tex]\(i^2 = -1\)[/tex], the expression becomes:
[tex]\[ \left(\frac{1}{2} + \frac{\sqrt{3}}{2} i \right) + \left(\frac{1}{2} i - \frac{\sqrt{3}}{2} \right) \][/tex]
Combine the real parts and the imaginary parts:
- Real parts: [tex]\(\frac{1}{2} - \frac{\sqrt{3}}{2}\)[/tex]
- Imaginary parts: [tex]\(\frac{\sqrt{3}}{2} i + \frac{1}{2} i = \left(\frac{\sqrt{3}}{2} + \frac{1}{2}\right) i\)[/tex]
So, the final expression is:
[tex]\[ \frac{1 - \sqrt{3}}{2} + \frac{1}{2} i \][/tex]
Thus, the real part of the expression is:
[tex]\[ \frac{1 - \sqrt{3}}{2} \approx -0.3660254037844386 \][/tex]
And the imaginary part is:
[tex]\[ \frac{1}{2} = 0.5 \][/tex]
Hence, the real part and the imaginary part are:
[tex]\[ \left(-0.3660254037844386, 0.5\right) \][/tex]
So, we have successfully determined the real and imaginary parts of the expression with the given values.
