https://file.aiquickdraw.com/mathbot/file/1730807842775n3bkpxua.jpg

The problem in "Aufgabe 2" requires converting the complex number \( z \) from polar to Cartesian coordinates. The polar coordinates given are:

- \( r = 5 \) (the modulus)
- \( \varphi = -\pi \) (the argument)

To convert from polar form \( z = r(\cos \varphi + i \sin \varphi) \) to Cartesian form \( z = x + iy \), we use:

\[
x = r \cos \varphi
\]
\[
y = r \sin \varphi
\]

1. **Calculate \( x \):**
   \[
   x = 5 \cos(-\pi) = 5 \cdot (-1) = -5
   \]

2. **Calculate \( y \):**
   \[
   y = 5 \sin(-\pi) = 5 \cdot 0 = 0
   \]

Thus, in Cartesian form, the complex number \( z \) is:
\[
z = -5 + 0i \quad \text{or simply} \quad z = -5
\]

Would you like further details or have any questions about this conversion process?

Here are some related questions that could deepen understanding:
1. How would the Cartesian form change if \( \varphi \) was \( \frac{\pi}{2} \)?
2. What is the interpretation of negative angles in polar coordinates?
3. How does changing \( r \) affect the location of \( z \) on the complex plane?
4. Can you convert other polar forms, such as \( r = 3 \) and \( \varphi = \frac{\pi}{4} \), to Cartesian coordinates?
5. What is the significance of the argument \( \varphi \) being exactly \( -\pi \)?

**Tip:** Remember that angles in polar form can often be expressed in multiple ways, as adding \( 2\pi \) (or subtracting) gives the same point on the complex plane.

Convert the complex number \( z \) from polar to Cartesian coordinates. Given polar coordinates are \( r = 5 \) and \( \varphi = -\pi \).

Learn how to convert the complex number \( z \) from polar coordinates (r = 5, φ = -π) to Cartesian coordinates. This example walks through using trigonometric conversions to determine the x and y values, resulting in the Cartesian form \( z = -5 + 0i \). Suitable for high school mathematics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation