Find the modulus-argument form of the complex number z=(5√ 3 - 5i)

The easiest way to complete questions of these types is to first sketch an Argand diagram. With 5√ 3 on the x (real) axis and -5 on the y (imaginary) axis, the modulus would be calculated simply by using pythagoras's theorem. Thus, the modulus of z would be equal to √((5√3)² + 5²) = √(75+25) = √100 = 10. The argument is then found as the angle between the real axis and the vector of the complex number. This can once again be calculated with trigonometry. As we know the magnitude of all three edges of the triangle, any of sin cos and tan operations can be used. In this example, i will compute it using tan. thus, tan(θ)=opp/adj = Imimaginary/real components = -5/(5√3) therefore arg(z)=arctan(-1/√3), which gives a value of -30° or -π/6 once we have both the modulus and the argument of the complex number, expressing it in modulus-argument form is straightforward. the complex number z= |z|((cos(arg(z) + isin(arg(z))) = 10(cos(-π/6) + isin(-π/6) ).