Math Problem Statement

Dame la respuesta correcta

e^(3π/2 * i) es igual a:

Solution

The expression given is e3π2ie^{\frac{3\pi}{2} i}. To solve this, we use Euler's formula:

eix=cos(x)+isin(x)e^{ix} = \cos(x) + i \sin(x)

Substituting x=3π2x = \frac{3\pi}{2}, we get:

e3π2i=cos(3π2)+isin(3π2)e^{\frac{3\pi}{2} i} = \cos\left(\frac{3\pi}{2}\right) + i \sin\left(\frac{3\pi}{2}\right)

Now calculate the values of cosine and sine at 3π2\frac{3\pi}{2}:

cos(3π2)=0\cos\left(\frac{3\pi}{2}\right) = 0 sin(3π2)=1\sin\left(\frac{3\pi}{2}\right) = -1

Thus:

e3π2i=0+i(1)=ie^{\frac{3\pi}{2} i} = 0 + i(-1) = -i

The correct answer is i\boxed{-i}, which corresponds to option b.

Would you like further details on this, or do you have any questions?

Here are 5 related questions for deeper understanding:

  1. How is Euler's formula applied in other complex exponentials?
  2. What is the geometric interpretation of eixe^{ix} on the complex plane?
  3. How does the unit circle relate to sine and cosine values for different angles?
  4. Can you derive the values of eiπe^{i\pi} and e2πie^{2\pi i} using Euler's formula?
  5. What are the properties of the imaginary unit ii?

Tip: Familiarize yourself with common angles like π/2,π,3π/2\pi/2, \pi, 3\pi/2, and 2π2\pi on the unit circle to quickly solve problems involving Euler's formula.

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Euler's Formula
Trigonometry

Formulas

Euler's Formula: e^(ix) = cos(x) + i sin(x)

Theorems

Euler's Formula

Suitable Grade Level

Grades 10-12