Simplifying a Complex Mathematical Expression Involving Trigonometry and Logarithms

Math Problem Statement

Simplify the expression: ((sqrt((13 * 5^2) * (2^4 + 1) + ln(e^(6i) + 3)))/2) * sin(90°).

Let's evaluate the expression given in the image:

$\frac{\sqrt{(18 \times 5^2) \times (2 + 1) + \ln(e^{6i} + 3)}}{2} \cdot \sin(90^\circ)$

We'll start by simplifying each component step-by-step:

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Trigonometry
Logarithms
Exponents
Complex Numbers

sin(90°) = 1
Square root
Natural logarithm ln(x)
Exponential function e^x

Euler's formula

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