If z = 4e^(iπ/3), find the rectangular form of z.

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Question: If z = 4e^(iπ/3), find the rectangular form of z.

Options:

Correct Answer: 2 + 2√3i

Solution:

z = 4(cos(π/3) + i sin(π/3)) = 4(1/2 + i√3/2) = 2 + 2√3i.

If z = 4e^(iπ/3), find the rectangular form of z.

If z = 4e^(iπ/3), find the rectangular form of z.

Step 1: Identify the given value of z, which is z = 4e^(iπ/3).
Step 2: Use Euler's formula, which states that e^(iθ) = cos(θ) + i sin(θ). Here, θ = π/3.
Step 3: Substitute θ into Euler's formula: e^(iπ/3) = cos(π/3) + i sin(π/3).
Step 4: Calculate cos(π/3) and sin(π/3). We find that cos(π/3) = 1/2 and sin(π/3) = √3/2.
Step 5: Substitute these values back into the equation: e^(iπ/3) = 1/2 + i(√3/2).
Step 6: Multiply the entire expression by 4: z = 4(1/2 + i√3/2).
Step 7: Distribute the 4: z = 4 * 1/2 + 4 * i√3/2.
Step 8: Calculate the products: z = 2 + 2√3i.
Step 9: Write the final answer in rectangular form: z = 2 + 2√3i.

Complex Numbers – Understanding the representation of complex numbers in polar and rectangular forms.
Euler's Formula – Using Euler's formula to convert between polar and rectangular forms of complex numbers.
Trigonometric Functions – Applying cosine and sine functions to find the real and imaginary parts of a complex number.