To convert the given complex number in polar form [tex]\( 4e^{i\frac{\pi}{15}} \)[/tex] into rectangular form, we need to follow a few key steps. We will express the complex number as [tex]\( z = a + bi \)[/tex], where [tex]\( a \)[/tex] is the real part and [tex]\( b \)[/tex] is the imaginary part.
The given complex number is of the form [tex]\( re^{i\theta} \)[/tex], where:
- [tex]\( r \)[/tex] is the magnitude of the complex number.
- [tex]\( \theta \)[/tex] is the phase angle.
For [tex]\( 4e^{i\frac{\pi}{15}} \)[/tex]:
- [tex]\( r = 4 \)[/tex]
- [tex]\( \theta = \frac{\pi}{15} \)[/tex]
We use the formula to convert from polar to rectangular form:
[tex]\[ z = r(\cos(\theta) + i\sin(\theta)) \][/tex]
Now, let's find the real and imaginary parts using the trigonometric functions cosine and sine.
1. Real Part:
[tex]\[
\text{Real Part} = r \cos(\theta)
\][/tex]
Substituting the values,
[tex]\[
\text{Real Part} = 4 \cos\left(\frac{\pi}{15}\right)
\][/tex]
2. Imaginary Part:
[tex]\[
\text{Imaginary Part} = r \sin(\theta)
\][/tex]
Substituting the values,
[tex]\[
\text{Imaginary Part} = 4 \sin\left(\frac{\pi}{15}\right)
\][/tex]
Thus, the rectangular form [tex]\( a + bi \)[/tex] of the complex number is given by:
[tex]\[ z = 4 \cos\left(\frac{\pi}{15}\right) + 4i \sin\left(\frac{\pi}{15}\right) \][/tex]
From the calculated values:
- [tex]\(\cos\left(\frac{\pi}{15}\right) \approx 0.9781476007338057\)[/tex]
- [tex]\(\sin\left(\frac{\pi}{15}\right) \approx 0.20791169081775931\)[/tex]
So we get:
- Real Part: [tex]\( 4 \times 0.9781476007338057 \approx 3.9125904029352228 \)[/tex]
- Imaginary Part: [tex]\( 4 \times 0.20791169081775931 \approx 0.8316467632710373 \)[/tex]
Therefore, the rectangular form of the given complex number [tex]\( 4e^{i\frac{\pi}{15}} \)[/tex] is:
[tex]\[ z \approx 3.9125904029352228 + 0.8316467632710373i \][/tex]
This is the complex number in rectangular form.
