z5=-3-4i

To solve for \( z_5 = -3 - 4i \), we can express this complex number in various forms, such as polar form, which can be helpful in various contexts such as finding roots or products of complex numbers.

Step 1: Rectangular Form

The given complex number is already in rectangular form:
\[ z_5 = -3 - 4i \]

Step 2: Polar Form

To convert to polar form, we need to find the magnitude \( r \) and the angle \( \theta \).

1. Magnitude \( r \): \[ r = |z_5| = \sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

2. Angle \( \theta \): The angle \( \theta \) can be found using the arctangent function. Since \( z_5 \) is in the third quadrant (both \( x \) and \( y \) are negative), we compute \( \theta \) as follows: \[ \theta = \tan^{-1} \left( \frac{-4}{-3} \right) = \tan^{-1} \left( \frac{4}{3} \right) \] To find the value in the third quadrant, we add \( \pi \): \[ \theta = \tan^{-1} \left( \frac{4}{3} \right) + \pi \]

   Numerically, using a calculator: \[ \tan^{-1} \left( \frac{4}{3} \right) \approx 0.9273 \] Thus, \[ \theta \approx 0.9273 + 3.1416 \approx 4.0689 \text{ radians} \quad (\text{approximately}) \]

3. Polar Form: Now we can express \( z_5 \) in polar form: \[ z_5 = r (\cos \theta + i \sin \theta) \] Therefore: \[ z_5 = 5 \left( \cos(4.0689) + i \sin(4.0689) \right) \]

Summary

• Rectangular Form: \( z_5 = -3 - 4i \)
• Magnitude: \( r = 5 \)
• Angle: \( \theta \approx 4.0689 \) radians
• Polar Form: \( z_5 = 5 \left( \cos(4.0689) + i \sin(4.0689) \right) \)

If you need to solve a specific problem involving this complex number, please let me know!