FMA 2 Complex numbers KUS objectives BAT know

FMA 2: Complex numbers • KUS objectives BAT know how to apply De. Moivres theorem to trigonometric identities Starter:

Notes 1 apply De Moivre’s theorem to trigonometric identities This involves changing expressions involving a function of θ into one without. For example changing a cos 6θ into powers of cosθ Remember n. Cr is a function you can find on your calculator The first term has the full power of n As you move across you slowly swap the powers over to the second term until it has the full power of n

WB 12 Express cos 3θ using powers of cosθ You have to think logically and decide where to start If we apply De Moivre’s theorem to this, we will end up with a ‘cos 3θ’ term If we apply the binomial expansion to it, we will end up with some terms with cosθ in So this expression is a good starting point! Apply De Moivre’s theorem Apply the Binomial expansion The two expressions we have made must be equal. Therefore the real parts in each and the imaginary parts in each must be the same Equate the real parts TA DA !We have successfully expressed cos 3θ as posers of cosθ!

we need something that will give us sin 6θ using De Moivre’s theorem Apply De Moivre’s theorem Apply the Binomial expansion This time we have to equate the imaginary parts as this has sin 6θ in 2 4

Notes 2 You also need to be able to work this type of question in a different way: Let: For example, you might have a power or cos or sin and need to express it using several linear terms instead Eg) Changing sin 6θ to sinaθ + sinbθ where a and b are integers To do this we need to know some other patterns first! We can add our two results together: We could also subtract our two results:

Notes 3 Taking this further Let: We could also subtract our two results: We could add our two results together:

WB 14 Express cos 5θ in the form acos 5θ + bcos 3θ + ccosθ Where a, b and c are constants to be found Creating a cos 5θ term Creating the other cos terms – use the Binomial expansion! These two expressions must be equal to each other TA DA! we have written cos 5θ using cos 5θ, cos 3θ and cosθ

Creating a sin 3θ term Creating the other sin terms – use the Binomial expansion! TA DA!

Creating a sin 4θ term Creating the cos terms – use the Binomial expansion! TA DA!

KUS objectives BAT know how to apply De. Moivres theorem to trigonometric identities self-assess One thing learned is – One thing to improve is –

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