If z = 2e^(iπ/3), find the rectangular form of z.

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Question: If z = 2e^(iπ/3), find the rectangular form of z.

Options:

Correct Answer: 1 + √3i

Solution:

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

If z = 2e^(iπ/3), find the rectangular form of z.

If z = 2e^(iπ/3), find the rectangular form of z.

Correct Answer: 1 + √3i

Step 1: Identify the given value of z, which is z = 2e^(iπ/3).
Step 2: Use Euler's formula, which states that e^(ix) = cos(x) + i sin(x).
Step 3: Substitute π/3 into 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 equation: 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).
Step 8: Simplify the expression: z = 1 + √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.
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