In modulus-argument form, z equals which expression?

In modulus-argument form, z equals which expression?

• A. z = r(cos θ + i sin θ)
• B. z = r(cos θ − i sin θ)
• C. z = r(cos θ) + i sin θ
• D. z = r(cos θ) + i r sin θ

Answer: (A)

In modulus-argument form, a complex number z is written using its magnitude r and angle θ as z = r (cos θ + i sin θ). This comes from z = x + i y with x = r cos θ and y = r sin θ, so the real part is r cos θ and the imaginary part is r sin θ. The plus sign inside the parentheses is essential because it gives the correct combination of real and imaginary parts scaled by the same magnitude r. If the imaginary part had a minus sign, it would correspond to the angle negated, not the standard modulus-argument form for θ. Writing the expression as r cos θ + i sin θ omits the r on the imaginary part, which would not reflect the full magnitude. The form r(cos θ) + i r sin θ is just another way to write the same thing as the standard form, since r distributes to both terms.

In modulus-argument form, a complex number z is written using its magnitude r and angle θ as z = r (cos θ + i sin θ). This comes from z = x + i y with x = r cos θ and y = r sin θ, so the real part is r cos θ and the imaginary part is r sin θ.

The plus sign inside the parentheses is essential because it gives the correct combination of real and imaginary parts scaled by the same magnitude r. If the imaginary part had a minus sign, it would correspond to the angle negated, not the standard modulus-argument form for θ. Writing the expression as r cos θ + i sin θ omits the r on the imaginary part, which would not reflect the full magnitude. The form r(cos θ) + i r sin θ is just another way to write the same thing as the standard form, since r distributes to both terms.