You're correct! Since the complex number $z = -1 + i$ lies in the second quadrant, the reference angle should indeed be $135^\circ$ (or $\frac{3\pi}{4}$ radians). Here's the correct reasoning:

Correcting the Argument Calculation

For the complex number $z = -1 + i$:

1. Calculate the reference angle:

   The basic angle (reference angle) associated with the ratio $\frac{1}{1}$ is $\tan^{-1}(1) = 45^\circ$ or $\frac{\pi}{4}$ radians.

2. Adjust for the quadrant:

   Since $z$ is in the second quadrant (where the real part is negative and the imaginary part is positive), the argument $\theta$ is given by:

   $\theta = 180^\circ - 45^\circ = 135^\circ$

   Or in radians:

   $\theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4}$

So the argument $\theta$ is indeed $135^\circ$ or $\frac{3\pi}{4}$ radians.

Final Answer

The complex number $z = \frac{4 + 6i}{1 - 5i}$ in:

• Trigonometric form: $\sqrt{2} \left( \cos\frac{3\pi}{4} + i\sin\frac{3\pi}{4} \right)$
• Exponential form: $\sqrt{2} e^{i\frac{3\pi}{4}}$

This confirms that the original calculation was correct in terms of the angle being $\frac{3\pi}{4}$ radians (or $135^\circ$) based on the quadrant.

If you have any more questions or need further clarification, feel free to ask!

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Here are some related questions you might find helpful:

1. How do you find the reference angle for complex numbers in other quadrants?
2. What is the significance of the reference angle in trigonometry?
3. How does the reference angle affect the signs of trigonometric functions in different quadrants?
4. Can you convert a complex number from trigonometric form back to rectangular form?
5. How do you compute powers of complex numbers using the exponential form?
6. What is the relationship between the argument of a complex number and its polar coordinates?
7. How does multiplying a complex number by $i$ affect its argument?
8. How do you add two complex numbers in trigonometric form?

Tip: Always double-check the quadrant when calculating the argument of a complex number to ensure the correct angle is used.