Let the point P represent the non-zero complex number Z=x+i y and directed line segment O P=r and it is inclined with positive direction of X - axis at an angle θ, then Complex number Z=x+i y in terms of r and θ can be written as

Question

Let the point P represent the non-zero complex number Z=x+i y and directed line segment O P=r and it is inclined with positive direction of X - axis at an angle θ, then Complex number Z=x+i y in terms of r and θ can be written as (A) Z=r( cos θ+i sin θ) (B) Z=r( sin θ+i cos θ) (C) Z=r( sin θ-i cos θ) (D) None of these

Tardigrade Admin · Accepted Answer

Let complex number z=x+i y ∵ O P=r, angle P O X=θ x=r cos θ ; y=r sin θ . Hence, z=r( cos θ+i sin θ) <img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/f86e89859392191fea6b9e66d6f9f012-.png" />

Q. Let the point P represent the non-zero complex number $Z = x + i y$ and directed line segment $OP = r$ and it is inclined with positive direction of $X$ - axis at an angle $θ$ , then
Complex number $Z = x + i y$ in terms of $r$ and $θ$ can be written as

$Z = r (cos θ + i sin θ)$

$Z = r (sin θ + i cos θ)$

$Z = r (sin θ - i cos θ)$

None of these

Let complex number $z = x + i y$
$∵ OP = r, ∠ POX = θ x = r cos θ; y = r sin θ .$
Hence, $z = r (cos θ + i sin θ)$

Q. Let the point P represent the non-zero complex number Z=x+iy and directed line segment OP=r and it is inclined with positive direction of X - axis at an angle θ, then Complex number Z=x+iy in terms of r and θ can be written as

Z=r(cosθ+isinθ)

Z=r(sinθ+icosθ)

Z=r(sinθ−icosθ)