Complex numbers A2

FMA2 Complex numbers Starter: KUS objectives BAT express a complex number in the form z = r(cosθ + isinθ) BAT express a complex number in the exponential form z = reiθ Starter:

By GCSE trigonometry, length x = rcosθ and length y = rsinθ Notes Modulus argument form of a complex number z (x,y) r x y θ You should remember the modulus-argument form of a complex number z = x + iy The value r is the modulus of the complex number, its distance from the origin (0,0) 𝑟= 𝑧 = 𝑥 2 + 𝑦 2 The argument is the angle the complex number makes with the positive x-axis, where: -π < θ ≤ π 𝑧=𝑥+𝑖𝑦 𝑧=𝑟𝑐𝑜𝑠𝜃 + 𝑖𝑟𝑠𝑖𝑛𝜃 By GCSE trigonometry, length x = rcosθ and length y = rsinθ 𝑧=𝑟(𝑐𝑜𝑠𝜃+𝑖𝑠𝑖𝑛𝜃) 𝑥=𝑟𝑐𝑜𝑠𝜃 𝑦=𝑟𝑠𝑖𝑛𝜃

cos(-θ) = cosθ sin(-θ) = -sinθ Notes 2 TRIG graphs C2 review y 90º 180º 270º -90º -180º -360º 1 -1 -270º θ y = cosθ You can see that cos(-θ) = cosθ anywhere on the graph -θ cos(-θ) = cosθ sin(-θ) = -sinθ 1 -1 90º 180º 270º y θ y = sinθ -θ You can see that sin(-θ) = -sinθ anywhere on the graph -90º -270º -360º -180º

WB 1 a) Express in the modulus-argument form: z = -√3 + i θ x y 1 This is the argument Start by sketching it on an Argand diagram Pay attention to the directions The ‘x’ part is negative so will go in the negative direction horizontally 𝑟= 3 2 + 1 2 = 2 𝑇𝑎𝑛𝜃= 1 3 𝜃= 𝜋 6 So 𝑧=2 𝑐𝑜𝑠 5𝜋 6 +𝑖𝑠𝑖𝑛 5𝜋 6 arg 𝑧= 5𝜋 6 Remember that the argument is not unique We could add 2π to them and the result would be the same, because 2π radians is a complete turn

𝑟= 1 2 + 1 2 = 2 𝑇𝑎𝑛𝜃= 1 1 𝜃= 𝜋 4 So 𝑧= 2 𝑐𝑜𝑠 − 𝜋 4 +𝑖𝑠𝑖𝑛 − 𝜋 4 WB 1 b) Express in the modulus-argument form: 𝑧= 1−𝑖 r 1 θ x y Start by sketching it on an Argand diagram Pay attention to the directions The ‘x’ part is negative so will go in the negative direction horizontally 𝑟= 1 2 + 1 2 = 2 𝑇𝑎𝑛𝜃= 1 1 𝜃= 𝜋 4 So 𝑧= 2 𝑐𝑜𝑠 − 𝜋 4 +𝑖𝑠𝑖𝑛 − 𝜋 4 arg 𝑧=− 𝜋 4 Remember that the argument is not unique We could add 2π to them and the result would be the same, because 2π radians is a complete turn

Notes 3 exponential form of a complex number If z = x + iy then the complex number can also be written in this way z = reiθ Later on you will meet series expansions of cosθ and sinθ. This can be used to prove the following result (which we will do when we come to chapter 6) As before, r is the modulus of the complex number and θ is the argument This form is known as the ‘exponential form’

start by sketching an Argand diagram and use it to find r and θ WB 2 a) Express the following complex number in the form reiθ, where -π < θ ≤ π z = 2 – 3i start by sketching an Argand diagram and use it to find r and θ r 2 θ x y 3 𝑟= 2 2 + (3) 2 = 13 𝑇𝑎𝑛𝜃= 3 2 𝜃=0.98 arg 𝑧=−0.98 𝑧= 𝑟𝑒 𝑖𝜃 𝑧= 13 𝑒 −0.98𝑖

WB 2 b) Express the following complex number in the form reiθ, where -π < θ ≤ π z = 2 𝑐𝑜𝑠 𝜋 10 +𝑖𝑠𝑖𝑛 𝜋 10 You can see from the form that r = √2 You can see from the form that θ = π/10 𝑟= 2 𝜃= 𝜋 10 𝑧= 𝑟𝑒 𝑖𝜃 𝑧= 2 𝑒 𝜋 10 𝑖 r 2 θ x y 3

z =5 𝑐𝑜𝑠 − 𝜋 8 +𝑖𝑠𝑖𝑛 − 𝜋 8 𝑧= 𝑟𝑒 𝑖𝜃 𝑧= 5𝑒 − 𝜋 8 𝑖 WB 2 c) Express the following complex number in the form reiθ, where -π < θ ≤ π z =5 𝑐𝑜𝑠 𝜋 8 −𝑖𝑠𝑖𝑛 𝜋 8 We need to adjust this first The sign in the centre is negative, we need it to be positive for the ‘rules’ to work We also need both angles to be identical. In this case we can apply the rules we saw a moment ago… Apply cosθ = cos(-θ) Apply sin(-θ) = -sin(θ) z =5 𝑐𝑜𝑠 − 𝜋 8 +𝑖𝑠𝑖𝑛 − 𝜋 8 r 2 θ x y 3 𝑧= 𝑟𝑒 𝑖𝜃 𝑧= 5𝑒 − 𝜋 8 𝑖

𝑧=𝑟(𝑐𝑜𝑠𝜃+𝑖𝑠𝑖𝑛𝜃) 𝑧= 2 𝑐𝑜𝑠 3𝜋 4 +𝑖𝑠𝑖𝑛 3𝜋 4 𝑧=−1+𝑖 WB 3 a) Express the following in the form, z = x + iy where 𝑥∈ℝ and 𝑦∈ℝ 𝑧= 2 𝑒 3𝜋 4 𝑖 You can see from the form that r = √2 You can see from the form that θ = 3π/4 𝑟= 2 𝜃= 3𝜋 4 𝑧=𝑟(𝑐𝑜𝑠𝜃+𝑖𝑠𝑖𝑛𝜃) 𝑥= 2 𝑟𝑐𝑜𝑠 3𝜋 4 𝑧= 2 𝑐𝑜𝑠 3𝜋 4 +𝑖𝑠𝑖𝑛 3𝜋 4 𝑦= 2 sin 3𝜋 4 𝑧=−1+𝑖 𝑥 2 + 𝑦 2 =2 𝑦 𝑥 = tan 3𝜋 4 = 1

WB 3 b) Express the following in the form, r(cosθ + isinθ), where 𝑥∈ℝ and 𝑦∈ℝ 𝑧=2 𝑒 23𝜋 5 𝑖 You can see from the form that r = 2 You can see from the form that θ = 23π/5 𝑟=2 𝜃= 23𝜋 5 𝑧=𝑟(𝑐𝑜𝑠𝜃+𝑖𝑠𝑖𝑛𝜃) 𝑧=2 𝑐𝑜𝑠 3𝜋 5 +𝑖𝑠𝑖𝑛 3𝜋 5 Subtract 2π 𝜃= 13𝜋 5 Subtract 2π 𝜃= 3𝜋 5

Use the relationships above to rewrite WB 4 Use: 𝑒 𝑖𝜃 =𝑐𝑜𝑠𝜃+𝑖𝑠𝑖𝑛𝜃 to show that: 𝑐𝑜𝑠𝜃= 1 2 𝑒 𝑖𝜃 + 𝑒 −𝑖𝜃 𝑧=𝑟(𝑐𝑜𝑠𝜃+𝑖𝑠𝑖𝑛𝜃) 𝑠𝑖𝑛 −𝜃 =−𝑠𝑖𝑛𝜃 𝑐𝑜𝑠 −𝜃 =𝑐𝑜𝑠𝜃 𝑧= 𝑟𝑒 𝑖𝜃 𝑒 𝑖𝜃 =𝑐𝑜𝑠𝜃+𝑖𝑠𝑖𝑛𝜃 Let θ = -θ 𝑒 𝑖(−𝜃) =𝑐𝑜𝑠 −𝜃 +𝑖𝑠𝑖𝑛 −𝜃 Use the relationships above to rewrite 𝑒 −𝑖𝜃 =𝑐𝑜𝑠𝜃−𝑖𝑠𝑖𝑛𝜃 𝑒 𝑖𝜃 =𝑐𝑜𝑠𝜃+𝑖𝑠𝑖𝑛𝜃 (1) 𝑒 −𝑖𝜃 =𝑐𝑜𝑠𝜃−𝑖𝑠𝑖𝑛𝜃 (2) Add 1 and 2 𝑒 −𝑖𝜃 + 𝑒 𝑖𝜃 =2𝑐𝑜𝑠𝜃 1 2 𝑒 −𝑖𝜃 + 𝑒 𝑖𝜃 =𝑐𝑜𝑠𝜃

Make sure you know the definition of 𝒆 −𝒊𝜽 Summary notes Summary points Make sure you know the definition of 𝒆 −𝒊𝜽 i e Which of the following is correct? 𝒆 −𝒊𝜽 =𝒄𝒐𝒔𝜽+𝒊 𝒔𝒊𝒏 𝜽 𝒆 −𝒊𝜽 =𝒔𝒊𝒏 𝜽+𝒊 𝒄𝒐𝒔 𝜽 𝒆 −𝒊𝜽 =𝒄𝒐𝒔 𝒊𝜽+𝒔𝒊𝒏 𝒊𝜽 𝒆 −𝒊𝜽 =𝒄𝒐𝒔 𝒊𝜽+𝒊 𝒔𝒊𝒏 𝒊𝜽 2. Subsequently, don’t get confused with the definition of 𝒆 𝒛 where z is a complex number 3. Make sure you can work comfortably with both the 𝒆 −𝒊𝜽 notation and the usual 𝒄𝒐𝒔𝜽+𝒊 𝒔𝒊𝒏 𝜽 Sometimes it is better to use the former and sometimes it’s better to use the latter, you need to be comfortable with both.

One thing to improve is – KUS objectives BAT express a complex number in the form z = r(cosθ + isinθ) BAT express a complex number in the exponential form z = reiθ self-assess One thing learned is – One thing to improve is –

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