Euler's relation and exponential form

Euler's relation

e^iθ = cos θ + i sin θ, with θ in radians.

One way to see why: put x = iθ in the Maclaurin series for e^x. The real terms give the series for cos θ and the imaginary terms give the series for sin θ.

With θ = π: e^iπ = -1.

Exponential form

A complex number with modulus r and argument θ is z = r(cos θ + i sin θ) = re^iθ.

The principal argument lies in -π < θ ≤ π. Always draw an Argand diagram to check the quadrant; the June 2025 examiners' report mentions confusion over the argument range and quadrant.

Multiplying and dividing

Index laws work: r1e^iα × r2e^iβ = r1r2 e^(i(α + β)) and r1e^iα ÷ r2e^iβ = (r1/r2) e^(i(α - β)).

Multiply the moduli, add the arguments. Adjust the final argument by ±2π to bring it into the principal range.

cos and sin as exponentials

e^-iθ = cos θ - i sin θ. Adding and subtracting:

cos θ = (e^iθ + e^-iθ)/2 and sin θ = (e^iθ - e^-iθ)/(2i).

These are used to prove trigonometric identities and lead to the z + 1/z methods later in this unit.

Worked example

z = 2e^iπ/3 and w = 3e^iπ/4. Find zw and z/w in exponential form.

1. zw: multiply the moduli, 2 × 3 = 6; add the arguments, π/3 + π/4 = 7π/12.
2. z/w: divide the moduli, 2/3; subtract the arguments, π/3 - π/4 = π/12.

Answer: zw = 6e^7πi/12 and z/w = (2/3)e^πi/12

Check yourself

1. Write 1 - i in exponential form.
   • √2 e^iπ/4
   • √2 e^-iπ/4
   • 2e^-iπ/4
   • √2 e^3iπ/4
   Show answer

   √2 e^-iπ/4. |1 - i| = √2. The point (1, -1) is in the fourth quadrant, so arg = -π/4.

2. Find the modulus of 2e^iπ/5 × (3e^iπ/7)².
   Show answer

   18. Moduli multiply: 2 × 3² = 18. The arguments do not affect the modulus.

3. z = e^iπ/6 and w = e^iπ/4. The argument of z³ w is kπ. Find k. Give your answer as a fraction in its simplest form.
   Show answer

   3/4. Arguments add: 3(π/6) + π/4 = π/2 + π/4 = 3π/4.

4. The principal argument of e^5iπ/3 × e^iπ/2 is kπ, with -1 < k ≤ 1. Find k. Give your answer as a fraction in its simplest form.
   Show answer

   1/6. 5π/3 + π/2 = 13π/6. Subtract 2π to get π/6.

5. Find the real part of 4e^2πi/3.
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   -2. 4(cos(2π/3) + i sin(2π/3)) = 4(-1/2) + 4i(√3/2). The real part is -2.

6. Which expression equals cos θ?
   • (e^iθ - e^-iθ)/2
   • (e^iθ - e^-iθ)/(2i)
   • (e^iθ + e^-iθ)/2
   • e^iθ + e^-iθ
   Show answer

   (e^iθ + e^-iθ)/2. Add e^iθ = cos θ + i sin θ and e^-iθ = cos θ - i sin θ: the sum is 2cos θ.

7. Evaluate e^iπ + 2.
   Show answer

   1. e^iπ = cos π + i sin π = -1, so the value is 1.

8. z = re^iθ. What is z × z*, where z* is the complex conjugate?
   • r² e^2iθ
   • r²
   • 2r cos θ
   • 1
   Show answer

   r². z* = re^-iθ, so zz* = r² e⁰ = r².

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3), Unit F2
• Pearson Edexcel IAL Further Pure Mathematics 2 scheme of work
• Pearson Edexcel IAL Further Pure Mathematics F2 (WFM02) examiners' report, June 2025

Last updated 8 November 2026. Spotted a mistake? Tell us.