In order to save energy, many, many things in nature move in a kind of repeated pattern we call a sine wave: water waves, sound waves, light waves, electricity, radio waves, and microwaves, for instance. The waves move in one direction quickly, then slow down until they stop, and begin moving the other way, faster and faster, and then they slow down again and change direction. If you throw a ball up into the air it will do this: fly up quickly, then slower, then fall slowly, then quickly into your hand. That’s gravity acting like a sine wave.

Tibetan singing bowl with sine waves in the water

The Indian mathematician Arya Bhata developed the idea of the sine function about 500 AD. He used math to describe this type of movement. The simplest sine function describes what happens to a right triangle when the other two angles change. The right angle is always 90 degrees, and the whole triangle is always 180 degrees, so the other two angles have to add up to 90 degrees. If one angle gets bigger, the other one has to get smaller. At the same time, the length of at least one of the sides will also have to change.

One way to describe these changes is to choose one of the two angles – A in this drawing – and divide the length of the side opposite that angle by the length of the hypotenuse. As the angle gets bigger, the opposite side will get longer, and the sine will get bigger. As the angle gets smaller, the opposite side will get smaller, and the sine will get smaller too.

Triangles and sines

If you make the angle A as big as 90 degrees it’s not a triangle but a straight line, and at more than 90 degrees, you’ve flipped your triangle over. If you draw a line representing what happens at all 360 degrees of angle A, you’ve got the sine wave in the middle of this page – the sine wave that represents how things like light, sound, electricity, and ocean waves move in nature.

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