Why is Trigonometry used in so many applications that have nothing to do with triangles?

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Why is Trigonometry used in so many applications that have nothing to do with triangles?

In: Mathematics

8 Answers

Anonymous 0 Comments

Everything, and I mean everything, can be broken down into triangles, yes, even circles. If you know your way around a triangle you can calculate anything.

Anonymous 0 Comments

While the origin of the word relates to angles and measures of a triangle, it is probably more useful to think of a triangle as the simplest polygon and that an “extreme” form of regular polygons, ie with the “most” sides, is the circle.

In that sense, the subject develops naturally to the study of circles and of particular importance in the real world, circular motion. With the further development in mathematics, it has been shown that any closed shape can be inscribed as a series of circular motions of circles of various shapes (although the number of circles might be very large to the extent of requiring an infinity of circles).

The other property of circular motion is that it is periodic (ie it eventually repeats). This is very useful for the analysis of periodic motion – the simplest being what many people think of as “waves”. Waves appear to be a very fundamental property of the universe (light waves, sound waves, radio waves etc etc) So the mathematics of trigonometry lead to circles and circular motion and then to waves and periodic motion in general. It becomes foundational to nearly all physics and engineering.

Anonymous 0 Comments

I am a mathematician. Triangles are used to calculate forces acting on objects at angles. In the simplest case, suppose you have an object sitting on an incline and want to calculate the rate of acceleration that it slides down the incline. Normal gravitational acceleration is 9.8m/s^2 however, the incline changes that. This is where the trig comes in, to account for the angle that the object is sliding down.

Anonymous 0 Comments

I’m not a mathematician. But I do know that triangles are the simplest shape (there’s no shape that has only two lines). Thus we can gather a lot about the ratios of lengths and angles and how they relate to one another from these very basic shapes.

These ratios also can form waves when plotted out on a graph. So we can use trigonometry to describe other things with waves, like sound waves and electromagnetic phenomena. Basic algebra functions give us lines and curves, but they don’t give us waves.

Anonymous 0 Comments

Because while initially trigonometry started off by using circles, we started defining it more generally using a unit circle. The x coordinate of a point on a unit circle is the cos and y coordinate is the sine. Since circles are the basis of the polar coordinate system, the math gets carried on too.

That’s one of the simple ways to explain it.

Anonymous 0 Comments

The first lie they teach you about trigonometry is that it comes from triangles. Trigonometry comes from properties of circles and the triangle stuff is just one useful application (if it came from triangles, then expressions like sin(190°) or cos(5pi/4) would not make any sense, since triangles can’t have angles that big). Circles in turn provide a very nice example of periodic motion: Going around a circle one and a half times puts me in the same place as going around it halfway, or two hundred sixty nine and a half times. If we for example want to understand something periodic but more complicated, then you can break it down as a sum of a bunch of sin and cos functions which is easier to analyse than a very complex expression.

Anonymous 0 Comments

Because really trigonometry is about circles, not triangles. It’s just you can use circles to prove things about trianges so it applies there.

Anonymous 0 Comments

Teaching cos and sin using triangles is common. But in my opinion, it’s actually a poor way of explaining these functions.

Draw a circle centered at (0, 0) with a radius of 1. Imagine you’re an ant walking around the circle at rate of 1 distance unit per second.

After t seconds of walking around the circle you’ll be at some position (x, y). The cos and sin functions are simply the x and y coordinates here.

That is, after say 0.37 seconds, the ant’s x position is cos(0.37) and its y position is sin(0.37).