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.
Six 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 six pages build them in that order, and every figure computes what it draws rather than loading a stored picture. Nearly all of them follow the angle you set. Two are deliberate stills: 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
Naming the sides, Pythagoras' theorem as a statement about areas, and the fact about similar triangles that makes a ratio depend on the angle alone.
The Unit Circle
Putting the triangle inside a circle of radius 1 so the definitions keep working past 90 degrees, and measuring angles in radians.
The Six Ratios as Lengths
Every one of the six functions drawn as a visible segment on that circle, which is where the names tangent, secant and the "co" prefix come from.
The Six Graphs
Unwrapping the circle into a wave, then reading period, range and asymptotes off each of the six curves.
The CAST Rule
Which functions are positive in which quadrant, worked out from the signs of x and y rather than memorised.
How They Are All Related
The three Pythagorean identities, the reciprocal and quotient identities, the exact special values, and where all of this gets used.
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.