Maths · Modulus and argument (or amplitude) of a complex number
If then arg
If \( z=\sqrt{\frac{1-i}{1+i}}, \) then arg \( z= \)
- A. \( \frac{\pi}{4}, \frac{\pi}{2} \)
- B. \( -\frac{\pi}{4}, \frac{\pi}{2} \)
- C. \( \frac{3 \pi}{4}, \pi \)
- D. \( -\frac{\pi}{4}, \frac{3 \pi}{4} \)
Step-by-step solution
First simplify (1-i)/(1+i) by multiplying numerator and denominator by (1-i): (1-i)^2/(1-i^2) = (1 - 2i + i^2)/2 = (-2i)/2 = -i. So z = √(-i). Write -i in polar form: modulus 1, argument -π/2 (or 3π/2). Then z = e^{i(-π/4)} and e^{i(3π/4)} (since square roots are ± e^{iθ/2}). Thus arguments are -π/4 and 3π/4.