Name: Applications of Complex Numbers (1.6.3) | DP IB Maths: AI HL Revision Notes 2021 Membership
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1.6.3 Applications of Complex Numbers

Frequency & Phase of Trig Functions

How are complex numbers and trig functions related?

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)

How are complex numbers used to add two sinusoidal functions?

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 + z₂ = ae^{i(bx + c)}+ de^{i(bx + e)} = e^ibx(ae^cⁱ+ de^eⁱ)
STEP 3: Convert ae^cⁱ+ de^eⁱinto 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

Exam Tip

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

Worked Example

Two AC voltage sources are connected in a circuit. If $V_{1} = 20 \sin (30 t)$ and $V_{2} = 30 \sin (30 t + 5)$ find an expression for the total voltage in the form $V = A \sin (30 t + B)$ .

1-6-3-ib-ai-hl-applications-of-complex-numbers-we-solution

DP IB Maths: AI HL

Revision Notes

1.6.3 Applications of Complex Numbers

Frequency & Phase of Trig Functions

How are complex numbers and trig functions related?

How are complex numbers used to add two sinusoidal functions?

Exam Tip

Worked Example

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