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If z = 2(cos(θ) + i sin(θ)), what is the value of |z|?

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Question: If z = 2(cos(θ) + i sin(θ)), what is the value of |z|?

Options:

  1. 2
  2. 4
  3. 1
  4. 0

Correct Answer: 2

Solution: 

|z| = 2, as |z| = r where z = r(cos(θ) + i sin(θ)).

If z = 2(cos(θ) + i sin(θ)), what is the value of |z|?

Practice Questions

Q1
If z = 2(cos(θ) + i sin(θ)), what is the value of |z|?
  1. 2
  2. 4
  3. 1
  4. 0

Questions & Step-by-Step Solutions

If z = 2(cos(θ) + i sin(θ)), what is the value of |z|?
Correct Answer: 2
  • Step 1: Identify the given expression for z, which is z = 2(cos(θ) + i sin(θ)).
  • Step 2: Recognize that the expression cos(θ) + i sin(θ) is in the form of Euler's formula, which represents a point on the unit circle.
  • Step 3: Note that in the expression z = r(cos(θ) + i sin(θ)), r is the magnitude (or modulus) of z.
  • Step 4: In this case, r is the coefficient in front of (cos(θ) + i sin(θ)), which is 2.
  • Step 5: Therefore, the magnitude |z| is equal to r, which is 2.
  • Complex Numbers – Understanding the representation of complex numbers in polar form and how to calculate their magnitude.
  • Magnitude of Complex Numbers – The magnitude of a complex number in polar form is given by the coefficient 'r' in the expression r(cos(θ) + i sin(θ)).
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