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Programme 2: Complex numbers 2 PROGRAMME 2 COMPLEX NUMBERS 2 STROUD Worked examples and exercises are in the text Programme 2: Complex numbers 2 Polar-form calculations Roots of a complex number Expansions Loci problems STROUD Worked examples and exercises are in the text Programme 2: Complex numbers 2 Polar-form calculations Roots of a complex number Expansions Loci problems STROUD Worked examples and exercises are in the text Programme 2: Complex numbers 2 Polar-form calculations Notation Positive angles Negative angles Multiplication Division STROUD Worked examples and exercises are in the text Programme 2: Complex numbers 2 Polar-form calculations Notation The polar form of a complex number is readily obtained from the Argand diagram of the number in Cartesian form. Given: z a jb then: r 2 a2 b2 so r a 2 b 2 and tan b so tan 1 b a a The length r is called the modulus of the complex number and the angle q is called the argument of the complex number STROUD Worked examples and exercises are in the text Programme 2: Complex numbers 2 Polar-form calculations Positive angles The shorthand notation for a positive angle (anti-clockwise rotation) is given as, for example: z r With the modulus outside the bracket and the angle inside the bracket. STROUD Worked examples and exercises are in the text Programme 2: Complex numbers 2 Polar-form calculations Negative angles The shorthand notation for a negative angle (clockwise rotation) is given as, for example: z r With the modulus outside the bracket and the angle inside the bracket. STROUD Worked examples and exercises are in the text Programme 2: Complex numbers 2 Polar-form calculations Multiplication When two complex numbers, written in polar form, are multiplied the product is given as a complex number whose modulus is the product of the two moduli and whose argument is the sum of the two arguments. If z1 r1 cos1 j sin 1 and z2 r2 cos 2 j sin 2 then z1z2 r1r2 cos1 2 j sin1 2 STROUD Worked examples and exercises are in the text Programme 2: Complex numbers 2 Polar-form calculations Division When two complex numbers, written in polar form, are divided the quotient is given as a complex number whose modulus is the quotient of the two moduli and whose argument is the difference of the two arguments. If z1 r1 cos1 j sin 1 and z2 r2 cos 2 j sin 2 then STROUD z1 r1 cos1 2 j sin1 2 z2 r2 Worked examples and exercises are in the text Programme 2: Complex numbers 2 Polar-form calculations Roots of a complex number Expansions Loci problems STROUD Worked examples and exercises are in the text Programme 2: Complex numbers 2 Polar-form calculations Roots of a complex number Expansions Loci problems STROUD Worked examples and exercises are in the text Programme 2: Complex numbers 2 Roots of a complex number De Moivre’s theorem nth roots STROUD Worked examples and exercises are in the text Programme 2: Complex numbers 2 Roots of a complex number De Moivre’s theorem If a complex number is raised to the power n the result is a complex number whose modulus is the original modulus raised to the power n and whose argument is the original argument multiplied by n. If z r cos j sin then z n r cos j sin = r n cos n j sin n n STROUD Worked examples and exercises are in the text Programme 2: Complex numbers 2 Roots of a complex number nth roots There are n distinct values of the nth roots of a complex number z. Each root has the same modulus and is separated from its neighbouring root by 2 radians n STROUD Worked examples and exercises are in the text Programme 2: Complex numbers 2 Polar-form calculations Roots of a complex number Expansions Loci problems STROUD Worked examples and exercises are in the text Programme 2: Complex numbers 2 Polar-form calculations Roots of a complex number Expansions Loci problems STROUD Worked examples and exercises are in the text Programme 2: Complex numbers 2 Expansions Trigonometric expansions Since: cos j sin n cos n j sin n then by expanding the left-hand side by the binomial theorem we can find expressions for: cos n and sin n in terms of powers of cos and sin STROUD Worked examples and exercises are in the text Programme 2: Complex numbers 2 Expansions Trigonometric expansions Let: z cos j sin then 1 cos j sin z so that: 1 z 2cos z zn 1 2cos n n z 1 z j 2sin z zn 1 j 2sin n n z from which we can expand cosn and sinn in terms of powers of cos and sin STROUD Worked examples and exercises are in the text Programme 2: Complex numbers 2 Polar-form calculations Roots of a complex number Expansions Loci problems STROUD Worked examples and exercises are in the text Programme 2: Complex numbers 2 Polar-form calculations Roots of a complex number Expansions Loci problems STROUD Worked examples and exercises are in the text Programme 2: Complex numbers 2 Loci problems The locus of a point in the Argand diagram is the curve that a complex number is constrained to lie on by virtue of some imposed condition. That condition will be imposed on either the modulus of the complex number or its argument. For example, the locus of z constrained by the condition that z 5 is a circle STROUD Worked examples and exercises are in the text Programme 2: Complex numbers 2 Loci problems The locus of z constrained by the condition that arg z is a straight line STROUD 4 Worked examples and exercises are in the text Programme 2: Complex numbers 2 Learning outcomes Use the shorthand form for a complex number in polar form Write complex numbers in polar form using negative angles Multiply and divide complex numbers in polar form Use de Moivre’s theorem Find the roots of a complex number Demonstrate trigonometric identities of multiple angles using complex numbers Solve loci problems using complex numbers STROUD Worked examples and exercises are in the text