Find the argument of the complex number z = -1 - i.

Find the argument of the complex number z = -1 - i.

Correct Answer: 3π/4

Step 1: Identify the complex number z = -1 - i. This means the real part is -1 and the imaginary part is -1.
Step 2: Use the formula for the argument θ of a complex number, which is θ = tan^(-1)(y/x), where y is the imaginary part and x is the real part.
Step 3: Substitute the values into the formula: θ = tan^(-1)(-1/-1).
Step 4: Simplify the fraction: -1/-1 = 1, so now we have θ = tan^(-1)(1).
Step 5: Find the angle whose tangent is 1. This angle is π/4 radians.
Step 6: Determine the correct quadrant for the angle. Since both the real part and imaginary part are negative, the complex number is in the third quadrant.
Step 7: In the third quadrant, the angle is π + π/4 = 5π/4.
Step 8: Therefore, the argument of the complex number z = -1 - i is θ = 5π/4.

Complex Numbers – Understanding the representation of complex numbers in the form z = 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 θ formed with the positive real axis in the complex plane, calculated using the arctangent function.
Quadrants in the Complex Plane – Recognizing the quadrant in which the complex number lies to determine the correct angle for the argument.