Products, quotients and powers in trigonometric form, and the exponential form
Week 5 · Products, quotients and powers in trigonometric form, and the exponential (Euler) form of a complex number · 3 lessons
3
Working in exponential form: products, quotients and back to x + yi
Pure Mathematics Sec 3 Term 1, Unit 2, Lesson 1
GoalMultiply, divide and raise to powers in exponential form, then return to x + yi.
New words
laws in exponential form · r1e^(iθ1) · r2e^(iθ2) = r1r2 e^(i(θ1+θ2)), and the quotient subtractsback to x + yi · re^(iθ) = r cos θ + i r sin θ
Explanation
In exponential form the rules of last lesson become the ordinary laws of exponents, which is why the notation is so useful: r1e^(iθ1) · r2e^(iθ2) = r1r2 e^(i(θ1 + θ2)), r1e^(iθ1) ÷ r2e^(iθ2) = (r1/r2) e^(i(θ1 − θ2)) and (re^(iθ))ⁿ = rⁿ e^(inθ). The table sets the two forms side by side. A good plan for a problem with several complex numbers: (1) write each number in exponential form with its modulus and principal amplitude; (2) apply the laws; (3) if the amplitude falls outside ]−π, π], add or subtract 2π; (4) if the question wants x + yi, finish with r cos θ + i r sin θ using the exact values. A quotient such as (1 + i)/(√3 − i) is much easier this way than by multiplying out with the conjugate, and the powers of a quotient are almost free.
Form
product z1·z2
quotient z1 ÷ z2
power zⁿ
trigonometric
r1r2 (cos(θ1+θ2) + i sin(θ1+θ2))
(r1/r2)(cos(θ1−θ2) + i sin(θ1−θ2))
rⁿ(cos nθ + i sin nθ)
exponential
r1r2 e^(i(θ1+θ2))
(r1/r2) e^(i(θ1−θ2))
rⁿ e^(inθ)
Worked examples
Compute (2e^(iπ/3))(3e^(iπ/2)) as x + yi. Moduli: 2 × 3 = 6. Amplitudes: π/3 + π/2 = 5π/6. So it is 6e^(5πi/6) = 6(−√3/2 + i/2) = −3√3 + 3i.
Simplify (1 + i)/(√3 − i). 1 + i = √2 e^(iπ/4) and √3 − i = 2e^(−iπ/6). The quotient is (√2/2) e^(i(π/4 + π/6)) = (√2/2) e^(5πi/12). Check by algebra: (1 + i)(√3 + i)/4 = ((√3 − 1) + (√3 + 1)i)/4, whose modulus is √(8)/4 = √2/2 and whose amplitude is 75° = 5π/12.
Find ((√3 + i)/(√3 − i))³. √3 + i = 2e^(iπ/6) and √3 − i = 2e^(−iπ/6), so the quotient is e^(iπ/3). Cubed: e^(iπ) = −1.
Write 2√2 e^(−3πi/4) as x + yi: 2√2(cos(−3π/4) + i sin(−3π/4)) = 2√2(−√2/2 − i√2/2) = −2 − 2i.
Class activity
Two ways. Compute (1 + i)(√3 + i) by ordinary multiplication, then by exponential form, and check that both give the same modulus 2√2 and angle 75°.
