Find the real part of the complex number z = 2e^(iπ/3).

Find the real part of the complex number z = 2e^(iπ/3).

Correct Answer: 1

Step 1: Identify the complex number given, which is z = 2e^(iπ/3).
Step 2: Recall that e^(iθ) can be expressed using Euler's formula: e^(iθ) = cos(θ) + i*sin(θ).
Step 3: Substitute θ with π/3 in Euler's formula: e^(iπ/3) = cos(π/3) + i*sin(π/3).
Step 4: Calculate cos(π/3) and sin(π/3). The values are cos(π/3) = 1/2 and sin(π/3) = √3/2.
Step 5: Substitute these values back into the expression: e^(iπ/3) = 1/2 + i*(√3/2).
Step 6: Multiply the entire expression by 2 (the coefficient in front of e): z = 2 * (1/2 + i*(√3/2)).
Step 7: Distribute the 2: z = 2 * 1/2 + 2 * i*(√3/2) = 1 + i√3.
Step 8: Identify the real part of the complex number z, which is 1.

Complex Numbers – Understanding the representation of complex numbers in exponential form and how to extract real and imaginary parts.
Euler's Formula – Using Euler's formula, e^(iθ) = cos(θ) + i*sin(θ), to convert complex exponentials to trigonometric form.
Trigonometric Values – Knowledge of basic trigonometric values, particularly for common angles like π/3.