We can simplify the given expression: 

\[
3.42 \cdot e^{-46.8i} - 3.7 \cdot e^{135i}
\]

This involves dealing with complex exponentials, which we can rewrite using Euler's formula:

\[
e^{ix} = \cos(x) + i \sin(x)
\]

### Step 1: Convert each term using Euler's formula.

For the first term:
\[
e^{-46.8i} = \cos(-46.8) + i \sin(-46.8) = \cos(46.8) - i \sin(46.8)
\]
Thus, 
\[
3.42 \cdot e^{-46.8i} = 3.42 \cdot (\cos(46.8) - i \sin(46.8))
\]

For the second term:
\[
e^{135i} = \cos(135) + i \sin(135)
\]
Thus,
\[
3.7 \cdot e^{135i} = 3.7 \cdot (\cos(135) + i \sin(135))
\]

### Step 2: Compute the trigonometric functions.

Let's compute these values and simplify the expression.import math

# Define the values for the trigonometric functions
angle1 = math.radians(46.8)
angle2 = math.radians(135)

# Compute cos and sin for both angles
cos_46_8 = math.cos(angle1)
sin_46_8 = math.sin(angle1)

cos_135 = math.cos(angle2)
sin_135 = math.sin(angle2)

# Calculate the complex expressions for each term
term1_real = 3.42 * cos_46_8
term1_imag = -3.42 * sin_46_8

term2_real = 3.7 * cos_135
term2_imag = 3.7 * sin_135

# Combine the real and imaginary parts
result_real = term1_real - term2_real
result_imag = term1_imag - term2_imag

result_real, result_imagThe simplified result of the expression is:

\[
4.96 - 5.11i
\]

This is the complex form after evaluating the given terms.

Would you like more details on the steps, or have any further questions?

Here are 5 related questions you might find interesting:
1. How do Euler's formulas help with complex exponentials?
2. What is the significance of the imaginary unit \(i\) in mathematics?
3. Can we convert this complex number into polar form?
4. How do trigonometric functions behave with negative angles?
5. How can complex numbers be visualized on the complex plane?

**Tip**: In complex numbers, the real part represents the x-axis, while the imaginary part corresponds to the y-axis on the complex plane.

Learn how to simplify complex exponentials using Euler's formula in the expression 3.42 e^-46.8i - 3.7 e^135i. This step-by-step guide covers converting exponentials to trigonometric form, calculating the trigonometric functions, and combining real and imaginary parts. Suitable for college-level students.

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