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# Exercise 2.7: Polar and Euler form of a Complex Number

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EXERCISE 2.7

1. Write in polar form of the following complex numbers

(i) 2 + i2ŌłÜ3

(ii) 3 - iŌłÜ3

(iii) -2 - i2

(iv) i -1 / [cos (ŽĆ/3) + i sin (ŽĆ/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 2╬Ė + i sin 2╬Ė , show that z = i tan╬Ė . 5. If cos ╬▒ + cos ╬▓ + cos ╬│ = sin ╬▒ + sin ╬▓ + sin ╬│ = 0, show that

(i) cos 3╬▒ + cos 3╬▓ + cos 3╬│ = 3cos(╬▒ + ╬▓ + ╬│ ) and

(ii) sin 3╬▒ + sin 3╬▓ + sin 3╬│ = 3sin (╬▒ + ╬▓ + ╬│ ) . 6. If z = x + iy and arg ( z-i  / z+2) = ŽĆ/4 , show that x2 + y2 + 3x - 3y + 2 = 0 .  