The t-formulas (AS)

The t-formulas are generally used to simplify or resolve trigonometric functions which are problematic to handle in their standard form. They involve replacing the variable in sine, cosine or tangent functions with a t-formula.
The three key t-formulas are given by the expressions: sinθ = 2t/(1 + t^2), cosθ = (1 - t^2)/(1 + t^2), and tanθ = sinθ/cosθ = 2t/(1 - t^2), with t = tan(θ/2).
To convert sin, cos, or tan functions into t-formulas, you set t = tan(θ/2). This results in simplified formulas that eliminate multi-angle formulas and allow simpler algebraic manipulations.
The t-formulas can be derived from the double-angle formulas in trigonometry. Double angle formulas arise from the compound angle formulas, and setting the two angles to be equal. For example, cos(2θ) simplifies to 1 - 2sin²θ or 2cos²θ - 1 or 1 - tan²θ/(1 + tan²θ).
One of the primary uses of the t-formulas is for integrating trigonometric functions. The t-formulas allow conversion of products of sines and cosines into a simple t-form, simplifying the integral considerably.
Additionally, they provide a means to simplify trigonometric expressions involving the sum of multiple angles.
T-formulas are also employed in finding particular solutions in trigonometric equations, especially where traditional methods prove challenging to apply.
When using the t-formulas, always be cautious about potentially missing one solution, because when you substituted θ/2 for t, your solution range effectively doubled.
Whenever solving trigonometric equations using t-formulas, remember to revert to the original variable after finding the solution in t-form.
It is essential to practise using the t-formulas across a range of problems to become comfortable with this powerful tool within the Further Pure Mathematics 1 syllabus.