. . De Moivre's Theorem De Moivre’s Theorem Given any complex number cos and any integer n , (cos sin ) Corollary ( ) (cos sin ) ( ) (cos sin ) ( ) (cos sin ) ( ) sin Now let us apply De Moivre's theorem to simplify complex numbers and to find solution of equations. De Moivre – Example .
If z , show that z Let z . By De Moivre's theorem , z n = cos z n = z Therefore, z + = cos + = 2cos n θ . Similarly, − = cos − = i sin θ . Example .
Simplify sin We have, sin = i . Raising to the power on both sides gives, = i i . Therefore, sin = . Example .
Simplify Let z = cos As | | z = | | z zz = , we get z Complex Numbers Therefore, = z z z . Therefore, = z = cos Example . Simplify (i) ( + i (ii) ( (i) ( + i Let + i = r . Then, we get r = ; tan , θ = ( + i lies in the first Quadrant) Therefore + i = Raising to power on both sides, + i By De Moivre's theorem, + i = cos = + i = ( ) i (ii) ( Let i = r .
Then, we get r = α = tan θ = ( i lies in II Quadrant) Therefore, i = Raising power on both sides, = = = = =