mathematics / trigonometry
Trigonometry
Six functions, one right triangle, and one circle. Sine, cosine, tangent, secant, cosecant and cotangent are all lengths you can point at, and Pythagoras' theorem is what ties them together.
Nine pages, in order
Trigonometry is usually handed over as a list of formulas. Six function names, three identities, a table of values, a mnemonic, and no explanation of where any of it came from. That is the wrong way round. All six functions come out of one right triangle, and everything that links them is Pythagoras' theorem applied to that triangle at different scales. These nine pages build them in that order, and every figure computes what it draws rather than loading a stored picture. The last three are the identities that combine two angles: the formulas for a sum or a difference of two angles, what those become when the two angles are equal, and how to turn a sum of two waves into a product. Nearly all of the figures follow the angles you set. Some are deliberate stills, among them the 3-4-5 counting figure on page 1 and the drawing of the touching line and the cutting line on page 3.
The Right Triangle
Where the six functions start. Name the sides, prove Pythagoras’ theorem by area, and see why scaling a triangle never changes its ratios.
The Unit Circle
The triangle definitions stop working past 90 degrees. Putting the triangle inside a circle of radius 1 fixes that, and turns sine and cosine into the coordinates of a point.
The Six Ratios as Lengths
Every one of the six functions is the length of a segment you can point at on the unit circle. That is where the names tangent, secant and the "co" prefix come from.
The Six Graphs
Unwrap the circle onto a horizontal axis and each function becomes a curve. Period, range and asymptotes are all readable straight off the circle that produced them.
The CAST Rule
Which functions are positive in which quarter of the circle, worked out from the signs of x and y rather than memorised, and how to get any angle’s value from an acute one.
How They Are All Related
Six functions, three identities and one table of exact values, all derived from a single right triangle with a hypotenuse of 1.
The Compound Angle Formulas
sin(A + B) is not sin A + sin B. What it is instead, proved once from a chord and once from a rectangle, and what it buys: adding two waves, and composing two rotations.
Double, Triple and Half Angles
What the compound angle formulas become when the two angles are the same: sin 2A, three forms of cos 2A, the half angle formulas read backwards, the triple angle polynomials, and the substitution that turns all six functions into fractions.
Sums into Products
Add two compound angle formulas together and half the terms cancel. What is left turns any sum of two sines or two cosines into a product, which is what makes an equation solvable in one line and what makes two out of tune strings wobble.