What is the real part of the complex number z = 2e^(iπ/3)?

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Question: What is the real part of the complex number z = 2e^(iπ/3)?

Options:

Correct Answer: 2

Solution:

The real part is 2 * cos(π/3) = 2 * 1/2 = 1.

What is the real part of the complex number z = 2e^(iπ/3)?

What is the real part of the complex number z = 2e^(iπ/3)?

Step 1: Identify the complex number 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. The real part is 1.

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