The t-formulae (Proof)

The t-formulae (Proof)

The t-formulae

Understanding the Basics

• The t-formulae (also called Euler’s Formulae) are a set of mathematical formulas that are used to simplify trigonometric functions by expressing them in terms of a variable t.

• These formulae can be particularly beneficial when solving integrals or derivatives that contain trigonometric functions.

The Formulae

• The t-formulae are as follows:

  • sinθ = 2t / (1 + t²)

  • cosθ = (1 - t²) / (1 + t²)

  • tanθ = 2t / (1 - t²)

  • where t = tan(θ/2)

Deriving the Formulae

• The t-formulae can be derived from basic trigonometric functions using substitution and the Pythagorean identity.

• For instance, the formula for sinθ can be derived by:

  • Letting t = tan(θ/2)

  • From the double angle formula, sinθ = 2tan(θ/2) / (1 + tan²(θ/2)), which simplifies to sinθ = 2t / (1 + t²) when t = tan(θ/2)

Proof of the Formulae

• To validate the t-formulae, we can employ the standard trigonometric identities:

  • cos²x + sin²x = 1 and tanx = sinx/cosx

• Substituting the t-formulae into these identities, they should be equivalent, which demonstrates the validity of the t-formulae.

Usage of the Formulae

• The t-formulae are a system of substitutions that facilitate the simplification of various trigonometric expressions.

• Specifically, they are often utilised in integration of trigonometric functions, where replacing sine, cosine, and tangent with the t-formulae can make the integral easier to solve.

Recall that the t-formulae can be a powerful tool when dealing with trigonometric functions in calculus. They simplify the process of integrating or differentiating trigonometric functions, so it’s essential to understand how and when to use them.