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!