Çözüldü Radical and Exponential Numbers - Principal Argument of Complex Numbers - The Pythagorean Theorem

A slightly hardened adaptation of an AYT practice problem for the SAT exam:

For a complex number whose real and imaginary parts are positive real numbers x and y, respectively,
if x^0.5·√(2y) = 32^(1 / 2) / 5 and √y / x^(1 / 2) = 3, then what is the principal argument of this number?

A) arctan(1 / 9)
B) arccot(4 / 15)
C) arcsin(15 / 36)
D) arccos(1 / √82)
E) arcsec(36 / √15)

2·x·y = 32 / 5 ⇒ x·y = 16 / 25....(I)
y / x = 9 ⇒ y = 9x....(II)
Substituting the equation in (II) for (I) yields x = 4 / 15....(III)
Moving the value in (III) to the equation (II) gives y = 36 / 15....(IV)
Based on the real and imaginary constants in (III) and (IV), the principal argument is arctan[ (36 / 15) / (4 / 15) ] = arctan(9 / 1), and applying the Pythagorean Theorem to the relevant right triangle, the hypotenuse is found to be (9^2 + 1^2)^0.5 = √82 units, so the principal argument is arccos(1 / √82).

Turkish Version of The Original Question: https://www.scribd.com/document/956271256/mert-hoca-ayt-deneme-1 (Page 1, Question 3)