Sin, cos and tan waves

ddx site catalogue - sin cos tan curvesFor many years I found those curves that appeared on the old oscilloscopes a slippery thing. How were they ‘sine waves’ and what on earth did it mean when people said that two wave forms were ’90 degrees out of phase’.

It helped when it first dawned on me what simple concepts sines and cosines actually were. Bit still..’90 degrees out of phase’?

So I like this design for its liveliness, for the neatness of the concept embodied in the small icons and, not least, because when I look at it sets my mind at rest.

I’ve given my amateur explanation of the nature of sines, cosines and the tangent elsewhere and I’ll assume you’ve either read it or don’t need to. In this design the meaning is embodied in the combination of the curves and the small icons.

Each curve is in effect the graph of a function – the two ‘waves’ are the sine and cosine functions with the colour dependent on which version of the design you’re looking at. The discontinuous curve is the tangent function. There is an imaginary vertical axis, and at the top of the design, where the waves peak, the value of the function is 1, at the bottom it is -1. Unsurprisingly, at the imaginary horizontal line through the middle of the design the value on the vertical axis is 0. The tangent regularly zooms off to infinity but we don’t have quite enough room to show that!

The imaginary horizontal axis of the graph consists of the angle of the point on the circumference of a circle where we’re measuring the sine and cosine. Since it’s imaginary we don’t have to specify whether we’re using degrees or radians to measure it but the value will be zero at each point where the curve of the sine function crosses the zero and 90° (or π/2) when the sine curve is at its peak.

The little circle icons label some significant points as we move round the circle –  namely angles at multiples of 45 degrees (or ¼ π radians). The dot on the circumference represents the angle from the centre, the two lines connecting the dot with the centre represent the value of the sine and cosine (matching the colour of the curves). When the value of the sine or cosine is 1 or -1, the relevant line in the little circle stretches all the way to the circumference. Since at that point the value of the other curve will always be 0, its length is represented by a dot of the relevant colour in the centre. When neither value is 1, the length of the lines indicate the relative values of the sine and cosine. Sine values above the centre point are positive, below it negative. Cosine value to the right of the centre point are positive, to the left negative.

So ’90 degrees out of phase’? Well if you look at the dots on two circles where sine and cosine curves are at maximum, you’ll see that the dot on the circumference is 90° difference. To get the two curves to superimpose and match perfectly (i.e. perfectly ‘in phase’), one of them has to be measured 90° later in the journey of the angle around the circle. Works for me.


Where mathematics becomes an object of desire.