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Find the rectangular form of the complex number if \(tan\theta = \frac{3}{4}\)1. z = 3 + 4i2. z = 4 + 3i3. z = 4 + 5i4. z = 4 - 3i |
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Answer» 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 |
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