The complex numbers and are given by: = 5 + ; = 2 3i z1 z2 z1 i z2 – Represent the points and on an Argand diagram z1 z2 Determine algebraically z1 z2 – Hence determine the modulus of z1 z2 – Determine a complex number in the form + where and are rational a bi, a b numbers. Hence, determine the argument of , giving your answer in radians correct to two decimal places. Determine the values of the real constants and such that: p q = 2i Trigonometric (polar) form of complex numbers From the above diagram: OP2 = = a2 + b2 r2 …………….[Pythagoras] r = And, tan = => = tan–1 From our earlier work of Trigonometry, we know that: sin = and cos = Thus, = sin and = cos b r a r = z a + bi = cos + ) z r r i(sin + )] z = r[cos i(sin Thus, if has a modulus and an argument , the polar form of is given by: z z = + i sin ] r[cos = cis
The complex numbers z1 and z2 are given by: z1 = 5 + i ; z2 = 2 – 3i Represent the points z1 and z2 on an Argand diagram Determine z1 – z2 algebraically Hence determine the modulus of z1 – z2 Determine a complex number in the form a + bi, where a and b are rational numbers. Hence, determine the argument of , giving your answer in radians correct to two decimal places. Determine the values of the real constants p and q such that: = 2i Trigonometric (polar) form of complex numbers From the above diagram: OP2 = r2 = a2 + b2 …………….[Pythagoras] r = And, tan = => = tan–1 From our earlier work of Trigonometry, we know that: sin = and cos = Thus, b = r sin and a = r cos z = a + bi z = r cos + r i(sin ) z = r[cos + i(sin )] Thus, if z has a modulus and an argument , the polar form of z is given by: = r[cos + i sin ] = cis
technical-mathematics-training-manual.docx