Write the complex number in exponential and trigonometric form z = 4+6i/1-5i

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 \( \theta \).

Learn how to express the complex number z = (4 + 6i) / (1 - 5i) in trigonometric and exponential forms. Follow step-by-step calculations to simplify the complex number, calculate its magnitude and argument, and convert it to trigonometric and exponential forms. Suitable for advanced high school students and those interested in complex numbers and trigonometry.

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