*the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse*in a right triangle (definition at wikipedia). The sine is also often represented as the ordinate of a point running on a unit circle centered at the origin. Any right triangle can be put in that setting. The length of the blue segment on Figure 1 illustrates this definition for an arbitrary angle α, in a circle of unit radius.

It is of interest to have in mind other geometric representations of the sine function. This helps to associate figures with sine and cosine operations or identities. The red segment in Figure 1 shows such an alternative representation, because the sine is also a chord of a circle of unit diameter. When we increment the angle by steps α, one endpoint of the chord advances on the little circle, while the other endpoint remains fixed at the origin, as illustrated by the animation in Figure 2.

When it represents the sine of angle α, the chord in the little circle spans an angle 2α. It is therefore a direct way to the angle multiplication and division by 2. Moreover, the endpoints of this chord can be placed two by two in any other direction on this circle. This enables one to find other variations on the recurrence pattern of the angle in the circle. During one of my circle drawing sessions, I was surprised by the stepwise zigzagging pattern of Figure 3.

Teaching the sine function with this zigzag pattern is particularly suited for daily life situations. Children like it, especially when you explain it at the table.

Hi Arjen,

ReplyDeleteYou are certainly an interesting father, as all I can remember hearing from my parents was not to play with my food :-) Anyway this zigzagging or oscillating pattern you’ve discovered hidden in the sine function is found also in mathematics as when calculating phi when using the Fibonacci sequence. That is if you look at how the sequence is approaching the solution to Phi it is also oscillating back and forth as the values diminish as it approachs.

Best,

Phil

Hi Phil,

ReplyDeleteI apologize for this long delay. I wanted to know a bit more about Fibonacci, his series and phi before responding but my attention got distracted with many other things:-)

The day when I wrote this post, I planned to eat pancakes. I usually draw that zigzagging pattern with whipped cream or maple syrup on a pancake. But due to different familial reasons, we chose to eat spaghetti instead, so I had to draw the zigzags with spaghetti and take the picture;-) Be assured, I won't do this in a restaurant. I'm much too conventionally minded;-)

I'll have to look better at Fibonacci's phi in order to understand its oscillation. Sideways, I learned that Fibonacci lived in the 13th century, that he called himself Leonardo Pisano. Only after his death, people called him Fibonacci. I took a look at his Book of Calculations and Book of Squares: tremendously interesting, how he gathered all these thoughts in the Middle Ages. He must have been a very original man.

Best wishes,

Arjen