t-Formulae and parameterisation of the circle

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,

t=\tan \frac{\theta}{2}

from which the following functions can be derived,

\tan \theta=\dfrac{2t}{1-t^{2}}\cos \theta =\dfrac{1-t^{2}}{1+t^{2}}\sin \theta=\dfrac{2t}{1+t^{2}}.

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:

t-formulae2

Euclid tells us that the angle subtended by the chord PP' at the centre is twice the angle subtended at the circumference.  The X-axis providing a line of symmetry, gives the relationship between the angles \frac{\theta}{2} and \theta at A and O respectively.

Defining t=\tan \theta and creating a line, y=t(x+1), through A with gradient t gives intersections with the Y-axis and the circle at R and P respectively.

The intersection point P can then be found by solving the simulataneous equations:

y=t(x+1), and, x^2+y^2=1.

Substituting for y leads to the quadratic,

(t^2+1)x^2+2t^2x+t^2-1=0,

which admits an easy factorisation once one acknowledges that it must have one root of x=-1.

(x+1)(x-\frac{1-t^2}{1+t^2})=0,

giving the other root, and x value for P as \frac{1-t^2}{1+t^2}.

Solving for y gives the co-ordinates of P.  When viewed as two altenative parameterisations of the unit circle, the derivation of the t formulae is complete.

\left(\dfrac{1-t^{2}}{1+t^{2}},\dfrac{2t}{1+t^{2}}\right)=(\cos \theta,\sin \theta ).