Question

Polar form of the complex number \(z = -4 + i4\sqrt{3}\) is :

• A. \(8(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3})\)
• B. \(8(\cos\frac{\pi}{6} + i\sin\frac{\pi}{6})\)
• C. \(8(\cos\frac{2\pi}{3} + i\sin\frac{2\pi}{3})\)
• D. \(8(\cos\frac{5\pi}{6} + i\sin\frac{5\pi}{6})\)

Hint:

We can quickly eliminate wrong options by checking the quadrant of the given complex number:
Since \(z = -4 + i4\sqrt{3}\) has a negative real part and a positive imaginary part, it MUST lie in the second quadrant.
Among the given angles:
- \(\frac{\pi}{3}\) and \(\frac{\pi}{6}\) are in the first quadrant.
- \(\frac{2\pi}{3}\) and \(\frac{5\pi}{6}\) are in the second quadrant.
Since \(\tan\alpha = \sqrt{3} \implies \alpha = 60^\circ\), the second quadrant angle is \(180^\circ - 60^\circ = 120^\circ = \frac{2\pi}{3}\).

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are given a complex number \(z = -4 + i4\sqrt{3}\) in rectangular form \(z = x + iy\).
We need to convert it into its equivalent polar form \(z = r(\cos\theta + i\sin\theta)\).
Key Formula or Approach:
1. Find the modulus \(r\):
\[ r = |z| = \sqrt{x^2 + y^2} \]
2. Find the argument \(\theta\):
First find the reference angle \(\alpha\) in the first quadrant:
\[ \tan\alpha = \left|\frac{y}{x}\right| \] Determine the quadrant of \(z = x + iy\):
- If \(x > 0, y > 0\) (Quadrant I): \(\theta = \alpha\)
- If \(x 0\) (Quadrant II): \(\theta = \pi - \alpha\)
- If \(x < 0, y < 0\) (Quadrant III): \(\theta = \alpha - \pi\)
- If \(x > 0, y < 0\) (Quadrant IV): \(\theta = -\alpha\)

Step 2: Detailed Explanation:

• Identify the real and imaginary parts of \(z = -4 + i4\sqrt{3}\):
\[ x = -4, \quad y = 4\sqrt{3} \]

• Calculate the modulus \(r\):
\[ r = \sqrt{(-4)^2 + (4\sqrt{3})^2} = \sqrt{16 + (16 \times 3)} = \sqrt{16 + 48} = \sqrt{64} = 8 \]

• Determine the reference angle \(\alpha\):
\[ \tan\alpha = \left|\frac{4\sqrt{3}}{-4}\right| = \sqrt{3} \] Since \(\tan\frac{\pi}{3} = \sqrt{3}\), we have:
\[ \alpha = \frac{\pi}{3} \]

• Determine the actual argument \(\theta\):
Since \(x = -4 < 0\) and \(y = 4\sqrt{3} > 0\), the complex number lies in the second quadrant.
Therefore:
\[ \theta = \pi - \alpha = \pi - \frac{\pi}{3} = \frac{2\pi}{3} \]

• Write \(z\) in polar form:
\[ z = r(\cos\theta + i\sin\theta) = 8\left(\cos\frac{2\pi}{3} + i\sin\frac{2\pi}{3}\right) \]

Step 3: Final Answer:
The polar form of the complex number is \(8\left(\cos\frac{2\pi}{3} + i\sin\frac{2\pi}{3}\right)\).