To express the complex number $z = \frac{4 + 6i}{1 - 5i}$ in exponential and trigonometric forms, we'll follow these steps:

Step 1: Simplify the Complex Number

First, let's simplify $z$ by multiplying the numerator and denominator by the conjugate of the denominator:

$z = \frac{4 + 6i}{1 - 5i} \times \frac{1 + 5i}{1 + 5i}$

$z = \frac{(4 + 6i)(1 + 5i)}{(1 - 5i)(1 + 5i)}$

The denominator simplifies using the difference of squares:

$(1 - 5i)(1 + 5i) = 1^2 - (5i)^2 = 1 - 25(-1) = 1 + 25 = 26$

Now, let's expand the numerator:

$(4 + 6i)(1 + 5i) = 4 \times 1 + 4 \times 5i + 6i \times 1 + 6i \times 5i$

$= 4 + 20i + 6i + 30i^2 = 4 + 26i + 30(-1) = 4 + 26i - 30 = -26 + 26i$

So, the complex number $z$ simplifies to:

$z = \frac{-26 + 26i}{26} = -1 + i$

Step 2: Convert to Trigonometric Form

The trigonometric form of a complex number $z = a + bi$ is given by:

$z = r(\cos \theta + i \sin \theta)$

Where $r$ is the magnitude (modulus) of $z$ and $\theta$ is the argument (angle) of $z$.

1. Calculate the magnitude $r$:

$r = \sqrt{a^2 + b^2} = \sqrt{(-1)^2 + 1^2} = \sqrt{1 + 1} = \sqrt{2}$

2. Calculate the argument $\theta$:

$\theta = \tan^{-1}\left(\frac{b}{a}\right) = \tan^{-1}\left(\frac{1}{-1}\right) = \tan^{-1}(-1)$

Since $z = -1 + i$ lies in the second quadrant (negative real part, positive imaginary part), the argument is:

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

Thus, the trigonometric form is:

$z = \sqrt{2} \left( \cos\frac{3\pi}{4} + i\sin\frac{3\pi}{4} \right)$

Step 3: Convert to Exponential Form

The exponential form of a complex number is:

$z = r e^{i\theta}$

Substituting the values we found:

$z = \sqrt{2} e^{i\frac{3\pi}{4}}$

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: $\sqrt{2} e^{i\frac{3\pi}{4}}$

Would you like further details or explanations?

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Here are some related questions to explore:

1. How do you multiply two complex numbers in exponential form?
2. What is the geometric interpretation of the argument $\theta$ in the complex plane?
3. How do you find the conjugate of a complex number in trigonometric form?
4. How does the magnitude of a complex number change under multiplication?
5. What is the relationship between polar and exponential forms of complex numbers?
6. How can you derive Euler's formula from the trigonometric form of a complex number?
7. What happens to the angle $\theta$ when you take the reciprocal of a complex number?
8. How do you solve complex equations involving both exponential and trigonometric forms?

Tip: When working with complex numbers, always pay attention to the quadrant in which the complex number lies to determine the correct argument $\theta$.