HSC · Mathematics Advanced · Functions · also HSC Trigonometric functions live from the app

Circular Functions and Their Graphs

Sine and cosine are the same wave shifted sideways, and every parameter in the equation maps to something you can see on the graph: amplitude is height, period is width, and the rest is position. This is the real knowscape from knowhere, not a picture of one. Drag it. Watch what actually changes.

mathematics advanced · functions · circular functions and their graphsdrag it · it is yours
the one idea

why this one carries the topic.

Every graph in this topic is one of a small handful of parent shapes, moved. y = af(n(x − h)) + k is a single sentence — stretch it, flip it, shift it — and it reads the same whether f is a sine wave or anything else you meet. Learn to read that sentence off a picture and write it back, and modelling periodic things stops being a new topic and becomes the one you already know.

Sine and cosine graphs are horizontally shifted versions of each other where amplitude stretches vertically, period compresses or stretches horizontally, and phase shift plus vertical translation position the wave in the plane.

what examiners catch — Students lose marks by confusing the period formula (period equals 2π divided by the coefficient of x, not the coefficient itself) and by forgetting that tan has period π while sin and cos have period 2π.
what you leave with

five things, not forty.

what's underneath

nothing here is a standalone fact.

knowhere maps every concept to what it rests on and what rests on it — 1 underneath this one, 1 built on top. Each one says why, in a sentence, not as an arrow on a diagram.

this conceptcircular functions and their graphsSine and cosine graphs are horizontally shifted versions of each other where amplitude stretches vertically, period compresses or stretches horizontally, and phase shift plus vertical translation position the wave in the plane.
sits under ittransformations of graphsbecause amplitude and period are transformations wearing trig clothes
built on itsolving trigonometric equationsbecause the answers live on the unit circle and its graph
the rest of functions

2 more, same treatment.

Each one is its own knowscape in the app — built for how a particular student takes things in, not one explanation handed to everybody.

modelling with functionsin the apptransformations of graphsin the app
this is one of 865

they don't read it. they run it.

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knowherehsc mathematics advancedcircular functions and their graphs