If z = 2 + 2i, what is the argument of z? (2023)

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Question: If z = 2 + 2i, what is the argument of z? (2023)

Options:

Correct Answer: π/4

Exam Year: 2023

Solution:

The argument of z is given by tan^(-1)(2/2) = tan^(-1)(1) = π/4.

If z = 2 + 2i, what is the argument of z? (2023)

If z = 2 + 2i, what is the argument of z? (2023)

Step 1: Identify the complex number z, which is given as z = 2 + 2i.
Step 2: Recognize that the argument of a complex number is the angle it makes with the positive x-axis in the complex plane.
Step 3: The real part of z is 2 and the imaginary part is also 2.
Step 4: Use the formula for the argument, which is tan^(-1)(imaginary part / real part).
Step 5: Substitute the values: tan^(-1)(2 / 2).
Step 6: Simplify the fraction: 2 / 2 = 1.
Step 7: Now calculate tan^(-1)(1).
Step 8: The angle whose tangent is 1 is π/4 radians.

Complex Numbers – Understanding the representation of complex numbers in the form a + bi, where a is the real part and b is the imaginary part.
Argument of a Complex Number – The argument of a complex number is the angle formed with the positive real axis in the complex plane, calculated using the arctangent of the ratio of the imaginary part to the real part.
Trigonometric Functions – Knowledge of how to use trigonometric functions, particularly the tangent and its inverse, to find angles.