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.
Functional notation and transformations is always tricky to teach and understand. GCSE students will meet this in Year 11. In general, transformations applied after the function are more easily understood:
Something very un-intuitive happens when the transformation is applied to the argument of the function:
with things stretching when they look like they should be compressing and other things moving the wrong way.
Mathematics is not always obvious, if it were we wouldn’t need it.
There’s always space on the inter-web for another proof of Pythagoras’s Theorem. Here’s one that uses the following equal areas property of parallelograms.
This kind of area chopping and shape translation is a feature of Euclidean geometry and our senses support it’s veracity at the order of size of the classroom.
The squares on the sides of a right-angle triangle set up a system of parallel lines which can then be used to demonstrate the Theorem using the above equal areas property.
The thread does not stop here though. Taking the parallel line structure which makes this work we get a generalisation of Pythagoras to non-right-angled triangles with the area of the parallelogram on the longest side being the sum of the areas of those constructed on the other two sides.
Bishop Berkeley writes this attack on the apparent supernatural reasoning involved in calculus. The infidel was probably Halley (of comet fame) or Newton.
If pupils find the subject difficult to understand at school, and teachers find it difficult to teach, then the reason may be articulated in this book by the great man.
“Now to conceive a Quantity infinitely small, that is, infinitely less than any sensible or imaginable Quantity, or any the least finite Magnitude, is, I confess, above my Capacity. But to conceive a Part of such infinitely small Quantity, that shall be still infinitely less than it, and consequently though multiply’d infinitely shall never equal the minutest finite Quantity, is, I suspect, an infinite Difficulty to any Man whatsoever”
“They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the Ghosts of departed Quantities?”
Some sympathy for the thesis is gained by Berkeley’s examination of tangent reasoning:
“Therefore the two errors being equal and contrary destroy each other; the first error of defect being corrected by a second error of excess. ……. If you had committed only one error, you would not have come at a true Solution of the Problem. But by virtue of a twofold mistake you arrive, though not at Science, yet at Truth. For Science it cannot be called, when you proceed blindfold, and arrive at the Truth not knowing how or by what means.”
The student of sixth form level mathematics who is eager to see how this is resolved must continue their path to mathematical enlightenment by studying the Analysis of Cauchy, Riemann and Weierstrass.
Logarithmic spirals frequently occur in nature. Is this such a manifestation?
The general equation for a logarithmic spiral is as follows.
Changing the variables and produces spirals of different qualities. The is really an enlargement scale factor but the controls how the spiral grows per revolution.
If is more representative of an ammonite then seems to be our Greenland depression.
Wouldn’t it be nice to take the equations for atmospheric fluid dynamics and show this explicitly. Unfortunately this is beyond the scope of this blog: here we have circular motion on the surface of a rotating sphere in an elliptical orbit … . The way to go with this would be to take the basic baratropic equations and then perform a scale analysis to disregard ‘small’ terms. At this point the pure mathematician get frustrated with approximations and goes into a sulk. We are left with contemplating the beautiful images though…
A level further mathematics question animated. Mid point of Normal x-intercept and Tangent y-intercept traces out another curve as original point moves round an ellipse.
Abstract algebra, the un-codified implications of which need to be fully understood at school to gain good grades in exams. School algebra and numeracy is all about persuading pupils that the numbers we have work by these rules, and not by the various coping patterns they learned earlier in their education.
FP3 2014 gives students the chance to prove the surface area of a sphere is 4 pi r squared. Interestingly four time the area of a cross section through the centre and the derivative of the volume. Pupils with strong maths general knowledge from GCSE would know what is coming. TOP TIP do as much maths challenge and extension work as possible, the general knowledge thus gained can only help. This FP3 2014 paper and many others are at my Exam Bucket: http://jped.co.uk/ExamBucket/exambucket.php.
Carr’s synopsis reveals much about cyclic quadrilaterals and the cosine rule. This Victorian text book is packed with sparse facts. Genius Ramanujan worked through these proving each one to himself. Modern students would extend their skills by doing the same. I have added some connections and associations to this sequence of facts to get you started. Should be within reach of year 11 pupils.