Edexcel IAL - Further Pure Mathematics 2- 3.1 Euler’s Formula- Study notes  - New syllabus

Euler’s Relation and Exponential Forms of Trigonometric Functions

Euler’s relation connects complex numbers, exponentials and trigonometric functions in one elegant formula.

Euler’s relation: \( e^{i\theta} = \cos\theta + i\sin\theta \)

This allows complex numbers to be written in exponential form and makes many calculations simpler.

Conjugate Form

Replacing \( \theta \) by \( -\theta \) gives:

\( e^{-i\theta} = \cos\theta – i\sin\theta \)

Adding and subtracting these two expressions gives important identities.

Expressions for Cosine and Sine

For cosine:

\( e^{i\theta} + e^{-i\theta} = (\cos\theta + i\sin\theta) + (\cos\theta – i\sin\theta) = 2\cos\theta \)

\( \cos\theta = \dfrac{1}{2}(e^{i\theta} + e^{-i\theta}) \)

For sine:

\( e^{i\theta} – e^{-i\theta} = (\cos\theta + i\sin\theta) – (\cos\theta – i\sin\theta) = 2i\sin\theta \)

\( \sin\theta = \dfrac{1}{2i}(e^{i\theta} – e^{-i\theta}) \)

Why Euler’s Relation is Useful

• It allows trigonometric identities to be proved algebraically.
• It simplifies multiplication and powers of complex numbers.
• It is used in differential equations and waves.