Use the trigonometric form of a complex number along with De

Use the trigonometric form of a complex number along with DeMoivre\'s theorem to calculate the given power. Write your answer in the standard form z = a + bi. (squareroot 3 + i)^2 z = __________

Solution

(sqrt3 + i)^2

I Z I= sqrt(sqrt3^2 + 1^2)=2

theta=tan^-1(1/sqrt3)= 30 degree

sqrt3+i=   2(cos30 + isin30)

(sqrt3 + 1)^2=(2(cos30 + isin30))^2

According to DEmoivre\'s theorem

(cos theta + isin theta)^n=cos n theta   + i sin n theta

Therefore (2(cos30+isin30))^2= 4(cos 60 + isin60)= 4(1/2 + isqrt3/2)

                                                                       = 2+2isqrt3