Correct Answer - Option 2 : z = 4 + 3i
CONCEPT:
If P represents the nonzero complex number z = x + iy.
Here \(r = \sqrt {{x^2} + {y^2}} = \left| z \right|\) is called modulus of given complex number.
CALCULATION:
Given: \(tanθ = \frac{3}{4}\:\)
As we know that, if z = x + iy then x = r cos θ and y = r sin θ
⇒ \(tan\ θ = \frac{y}{x}\:\)
⇒ \(tanθ = \frac{3}{4}\:= \frac{y}{x}\)
\(\therefore r = \sqrt {{{\left( x \right)}^2} + {{\left( y \right)}^2}} = \sqrt {{{\left( 4 \right)}^2} + {{\left( 3 \right)}^2}} = 5\)
\( ⇒ cosθ = \frac{x}{r} = \frac{4}{5};\;sinθ = \frac{y}{r} = \frac{3}{5}\)
\(\therefore z = r\left( {cosθ + i\;sinθ } \right) = 5\left( {\frac{4}{5} + \frac{3}{5}i} \right)\)
⇒ z = 4 + 3i
