Exercise 2.7: Polar and Euler form of a Complex Number

EXERCISE 2.7

1. Write in polar form of the following complex numbers

(i) 2 + i2√3

(ii) 3 - i√3

(iii) -2 - i2

(iv) i -1 / [cos (Ï€/3) + i sin (Ï€/3)].

2. Find the rectangular form of the complex numbers

3. If ( x1 + iy1 )( x2 + iy2 )( x3 + iy3 )... ...( xn + iyn ) = a + ib , show that

(i) (x12 + y12 )(x22 + y22 )(x32 + y32 )... ...(xn2 + y n2 ) = a2 + b2

(ii) 

4. If 1+ z / 1- z = cos 2θ + i sin 2θ , show that z = i tanθ .

5. If cos Î± + cos Î² + cos Î³ = sin Î± + sin Î² + sin Î³ = 0, show that

(i) cos 3α + cos 3β + cos 3γ = 3cos(α + β + γ ) and

(ii) sin 3α + sin 3β + sin 3γ = 3sin (α + β + γ ) .

6. If z = x + iy and arg ( z-i  / z+2) = Ï€/4 , show that x2 + y2 + 3x - 3y + 2 = 0 .