To find the real part of the complex number \(75e^{-1.49i}\), we can use Euler's formula:

\[
z = re^{i\theta} = r(\cos\theta + i\sin\theta)
\]

where:
- \(r = 75\)
- \(\theta = -1.49\)

The real part of \(z\) is given by:

\[
\text{Re}(z) = r \cos(\theta)
\]

Substituting the values:

\[
\text{Re}(75e^{-1.49i}) = 75 \cos(-1.49)
\]

Since \(\cos(-\theta) = \cos(\theta)\), this simplifies to:

\[
\text{Re}(75e^{-1.49i}) = 75 \cos(1.49)
\]

Now let's calculate this value.The real part of \(75e^{-1.49i}\) is approximately **6.05**.

Would you like more details on this calculation or have any questions?

Here are 5 related questions that expand on this topic:
1. What is the imaginary part of \(75e^{-1.49i}\)?
2. How do you convert a complex number from polar form to rectangular form?
3. What is Euler's formula, and how is it used to represent complex numbers?
4. How would you find the magnitude and argument of a complex number given in rectangular form?
5. What are some real-life applications of complex numbers?

**Tip:** Remember that the cosine function is periodic, so understanding its properties can help in solving more advanced problems involving complex numbers.

Discover how to calculate the real part of the complex number 75e^-1.49i using Euler's formula. This problem covers the use of trigonometric functions to find the real part and involves key concepts in complex numbers, suitable for high school students in Grades 10-12.

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