Key Concepts and Formulas

• Complex Conjugate Properties: $\overline{\overline{z}} = z$, $\overline{z_1 + z_2} = \overline{z_1} + \overline{z_2}$, $\overline{iz} = -i\overline{z}$
• Argument Properties: $\arg(z_1 z_2) = \arg(z_1) + \arg(z_2)$, $\arg(\frac{z_1}{z_2}) = \arg(z_1) - \arg(z_2)$, $\arg(z^n) = n \arg(z)$
• Argument of i: $\arg(i) = \frac{\pi}{2}$

Step-by-Step Solution

1. Express z in terms of w using the given condition.

We are given $\overline z + i\overline w = 0$. Our goal is to isolate $\overline{z}$ and then eliminate the conjugates to find a direct relationship between $z$ and $w$.

Isolating $\overline{z}$, we have: $\overline z = -i\overline w$

Now, take the conjugate of both sides. This is done to eliminate the conjugate notation. $\overline{\overline z} = \overline{-i\overline w}$

Applying conjugate properties: $z = \overline{-i} \cdot \overline{\overline{w}}$ $z = i w$

2. Express w in terms of z, then substitute into the second given equation.

From the previous step, we have $z = iw$. Solving for $w$, we get: $w = \frac{z}{i}$

We are also given $\arg(zw) = \pi$. We want to substitute our expression for $w$ in terms of $z$ to get an equation involving only $z$.

Substituting $w = \frac{z}{i}$ into $\arg(zw) = \pi$, we get: $\arg\left(z \cdot \frac{z}{i}\right) = \pi$ $\arg\left(\frac{z^2}{i}\right) = \pi$

3. Apply argument properties to simplify the equation.

We use the property $\arg(\frac{z_1}{z_2}) = \arg(z_1) - \arg(z_2)$: $\arg(z^2) - \arg(i) = \pi$

Then, we use the property $\arg(z^n) = n \arg(z)$: $2\arg(z) - \arg(i) = \pi$

4. Solve for arg(z).

We know that $\arg(i) = \frac{\pi}{2}$. Substituting this into the equation: $2\arg(z) - \frac{\pi}{2} = \pi$

Adding $\frac{\pi}{2}$ to both sides: $2\arg(z) = \pi + \frac{\pi}{2} = \frac{3\pi}{2}$

Dividing both sides by 2: $\arg(z) = \frac{3\pi}{4}$

Common Mistakes & Tips

• Remember that $\overline{i} = -i$. Failing to apply this correctly is a common source of error.
• Be careful with the signs when applying argument properties, especially when dealing with division.
• Always double-check your final answer to make sure it makes sense within the context of the problem.

Summary

By using the properties of complex conjugates and arguments, we were able to simplify the given equations and solve for $\arg(z)$. We first expressed $z$ in terms of $w$ and then substituted this into the second equation to eliminate $w$. Then, using argument properties, we simplified and solved for $\arg(z)$. The final answer is $\frac{3\pi}{4}$.

Final Answer The final answer is $\boxed{\frac{3\pi}{4}}$, which corresponds to option (C).