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If z = 2e^(iπ/3), what is the value of z?

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Question: If z = 2e^(iπ/3), what is the value of z?

Options:

  1. 1 + i√3
  2. 2 + 0i
  3. 0 + 2i
  4. 2 - 2i

Correct Answer: 1 + i√3

Solution: 

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

If z = 2e^(iπ/3), what is the value of z?

Practice Questions

Q1
If z = 2e^(iπ/3), what is the value of z?
  1. 1 + i√3
  2. 2 + 0i
  3. 0 + 2i
  4. 2 - 2i

Questions & Step-by-Step Solutions

If z = 2e^(iπ/3), what is the value of z?
  • 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: 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 + i√3.
  • Complex Numbers – Understanding the representation of complex numbers in polar form and converting them to rectangular form using Euler's formula.
  • Trigonometric Functions – Applying cosine and sine values for specific angles to compute the real and imaginary parts of a complex number.
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