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IB DP Maths: AI HL復習筆記1.6.3 Applications of Complex Numbers

Category: IB課程, 教材筆記, 福利干貨 Date: 2022年7月20日上午10:45

A sinusoidal function is of the form?a?sin(bx?+?c)
- a?represents the?amplitude
- b?represents the?period?(also known as frequency)
- c?represents the?phase shift
  - The function may be written?a?sin(bx?+?bc) =?a?sinb(x?+?c) where the phase shift is represented by?bc
  - This will be made clear in the exam
When written in?modulus-argument?form the?imaginary part?of a complex number relates only to the?sin?part and the?real?part relates to the?cos?part
- This means that the complex number can be rewritten in Euler's form and relates to the sinusoidal functions as follows:
- a?sin(bx?+?c) = Im (ae^{i(bx?+ c)})
- a?cos(bx?+?c) = Re (ae^{i(bx?+ c)})
Complex numbers are particularly useful when working with electrical currents or voltages as these follow sinusoidal wave patterns
- AC voltages may be given in the form V =?a?sin(bt?+?c) or V =?a?cos(bt?+?c)

Complex numbers can help to add two?sinusoidal?functions if they have the same?frequency?but different?amplitudes?and?phase shifts
- e.g. 2sin(3x?+ 1) can be added to 3sin(3x?- 5) but?not?2sin(5x?+ 1)
To add?asin(bx?+?c) to?dsin(bx?+?e)
- or?acos(bx?+?c) to?dcos(bx?+?e)
STEP 1: Consider the complex numbers?z₁?=?ae^{i(bx?+ c)}?and?z₂?=?de^{i(bx?+ e)}
- Then?asin(bx?+?c) +?dsin(bx?+?e) = Im (z₁?+?z₂)
- Or?acos(bx?+?c) +?dcos(bx?+?e) = Re (z₁?+?z₂)
STEP 2: Factorise?z₁?+?z_2??=?ae^{i(bx?+ c)}_?+?de^{i(bx?+ e)}?= e^ibx?(ae^c^i?+?de^eⁱ)
STEP 3: Convert?ae^c^i?+?de^e^i??into a single complex number in exponential form
- You may need to convert it into Cartesian form first, simplify and then convert back into exponential form
- Your GDC will be able to do this quickly
STEP 4: Simplify the whole term and use the rules of indices to collect the powers
STEP 5: Convert into polar form and take...
- only the imaginary part for sin
- or only the real part for cos

An exam question involving applications of complex numbers will often be made up of various parts which build on each other
- Remember to look back at your answers from previous question parts to see if they can help you, especially when looking to convert from Euler's form to a sinusoidal graph form