To solve the problem of expressing [tex]\(\left[4\left(\cos \frac{\pi}{10} + i \sin \frac{\pi}{10}\right)\right]^5\)[/tex] in both rectangular form [tex]\(x + yi\)[/tex] and exponential form [tex]\(r e^{i \theta}\)[/tex], we can use De Moivre's Theorem and polar to rectangular conversions.
### Step-by-Step Solution:
1. Convert to Polar Form:
The given expression is already in polar form [tex]\[ r \left( \cos \theta + i \sin \theta \right) \][/tex] where:
- [tex]\(r = 4\)[/tex]
- [tex]\(\theta = \frac{\pi}{10}\)[/tex]
2. Apply De Moivre's Theorem:
De Moivre's Theorem states that for a complex number in polar form [tex]\(r \left( \cos \theta + i \sin \theta \right)\)[/tex], raising it to the power of [tex]\(n\)[/tex] results in:
[tex]\[
\left[ r \left( \cos \theta + i \sin \theta \right)\right]^n = r^n \left( \cos (n\theta) + i \sin (n\theta) \right)
\][/tex]
Here, we have [tex]\(n = 5\)[/tex].
3. Calculate the Magnitude:
The magnitude [tex]\(r^n\)[/tex] is:
[tex]\[
r^n = 4^5 = 1024
\][/tex]
4. Calculate the Angle:
The angle [tex]\(n \theta\)[/tex] is:
[tex]\[
n \theta = 5 \left( \frac{\pi}{10} \right) = \frac{5\pi}{10} = \frac{\pi}{2}
\][/tex]
5. Convert to Rectangular Form:
In rectangular form, we have:
[tex]\[
x + yi = 1024 \left( \cos \frac{\pi}{2} + i \sin \frac{\pi}{2} \right)
\][/tex]
Using the unit circle values:
- [tex]\(\cos \frac{\pi}{2} = 0\)[/tex]
- [tex]\(\sin \frac{\pi}{2} = 1\)[/tex]
Thus:
[tex]\[
x + yi = 1024 \left( 0 + i \cdot 1 \right) = 0 + 1024i
\][/tex]
Therefore, the rectangular form is:
[tex]\[
0 + 1024i \quad \text{or} \quad 1024i
\][/tex]
- [tex]\( x = 6.270191611634448 \times 10^{-14} \)[/tex] (very close to 0 due to computational precision)
- [tex]\( y = 1024.0 \)[/tex]
6. Express in Exponential Form:
The exponential form is:
[tex]\[
r e^{i \theta} = 1024 e^{i \frac{\pi}{2}}
\][/tex]
### Final Answer:
- Rectangular form: [tex]\(6.270191611634448 \times 10^{-14} + 1024i\)[/tex]
- Exponential form: [tex]\(1024 e^{i \frac{\pi}{2}}\)[/tex]
