Question

find the modulus and amplitude of the following complex number
-4 - 4i​

Answer 1

Step-by-step explanation:

If z = a + ib is any complex number such that a ∈R, b∈R  

and i = √-1,

then the modulus of complex number, |Z| is given by

|Z| = √a² + b²

Amplitude of the complex number , θ is given by,

θ = tan⁻¹(b/a)

i) Z = 4 - 4i

Modulus,|Z| = √4² + 4² = √32

Amplitude,θ = tan⁻¹(-4/4) = -tan⁻¹(4/4)

i

Answer 2

Answer:

$\huge\underline{ \underline{ \red {your \: \: \: answer}}}$

$\bf\red{ \boxed{ \bigstar\mapsto: modulas |z| = \sqrt{ = 2} \bigstar}} \\ \bf\pink{ \boxed{ \bigstar\mapsto: its \: amplitude =\frac{\pi}{4} \bigstar }}$

Step-by-step explanation:

$\bf\blue{ \boxed{complex \: number }} \\ \\ \bf\red{ \mapsto: z = - 4 - 4i} \\ \bf\green{ \mapsto:z = a + ib } \\ \bf\orange{ \mapsto: |z| = \sqrt{ {a}^{2} + {b}^{2} } } \\ \\ \bf\purple{\therefore |z| = \sqrt{( { - 4}^{2} ) + ( { - 4}^{2} )} } \\ \\ \bf\green{ \boxed{ \bigstar\implies: \sqrt{32} = 4 \sqrt{2} \: \: \bigstar } } \\ \\ \bf\red{ \mapsto:here } \\ \\ \bf\pink{x = - 4 \: \: \: \: y = - 4} \\ \\ \bf\green{ \boxed{ ampitude = {tan}^{ - 1} (\frac{y}{x} )} }\\ \\ \bf\orange{ So, \: the \: modulus \: of \: the \:} \\ \bf \orange{given \: complex \: number \: is \mapsto} \\ \bf\pink{ \implies \theta = {tan}^{ - 1}( \frac{ - 4}{ - 4} ) } \\ \\ \bf\green{ \boxed{ \bigstar\implies \theta = \frac{ \pi}{4} \: \: \bigstar} } \\ \\ \large\boxed{ \fcolorbox{violet}{lime}{hope \: it \: help \: you}}$