1. Tardigrade
  2. Question
  3. Mathematics
  4. Value of (( cos θ+i sin θ)4/( cos θ-i sin θ)3) is

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. Value of $\frac{(\cos \theta+i \sin \theta)^{4}}{(\cos \theta-i \sin \theta)^{3}}$ is

Complex Numbers and Quadratic Equations

Solution:

$\frac{(cos\, \theta+i \,sin \,\theta)^{4}}{(cos\, \theta-i \,sin\, \theta)^{3}}$
$=(cos\, \theta+i \,sin\, \theta)^{4}(cos \,\theta-i \,\sin\, \theta)^{-3}$
$=(cos \,4 \,\theta+ i \,sin \,4 \theta)\{cos (-\theta)+ i \,sin (-\theta)\}^{-3}$
$=(cos \,4 \,\theta+ i \,sin\, 4 \theta)\{cos (-3)(-\theta)+ i\,sin (-3)(-\theta)\}$
$=(cos \,4 \theta+ i \,sin \,4 \theta)\{cos \,3 \theta+ i \,sin \,3 \theta\}$
$=cos \,4 \theta\, cos \,3 \theta-sin \,4 \theta \,sin \,3 \theta$
$+ i (sin \,4 \theta\,cos \,3 \theta+sin \,3 \theta\,cos \,4 \theta)$
$=cos (4 \theta+3 \theta)+ i\, sin (4 \theta+3 \theta)=cos\, 7 \theta+ i \,sin \,7 \theta$