t-Formulae are used in integration to tackle rational expressions of tigonometric functions. After a spell in the cold, when they were not included in some A level specifications, they are now back in sixth form lessons.
It all starts with the subsitution,
from which the following functions can be derived,
, , .
These derivations can be made using compound trigonmetry fomulae. Alternatively, there is an engaging co-ordinate geometry derivation which has the merits of doubling up as an algebraic parametrisation of the circle.
Start with the following figure:
Euclid tells us that the angle subtended by the chord at the centre is twice the angle subtended at the circumference. The -axis providing a line of symmetry, gives the relationship between the angles and at and respectively.
Defining and creating a line, , through with gradient gives intersections with the -axis and the circle at and respectively.
The intersection point can then be found by solving the simulataneous equations:
, and, .
Substituting for leads to the quadratic,
which admits an easy factorisation once one acknowledges that it must have one root of .
giving the other root, and value for as .
Solving for gives the co-ordinates of . When viewed as two altenative parameterisations of the unit circle, the derivation of the formulae is complete.