# How did people use trigonometric functions without calculators or other devices?

#### MegaMoh

##### New member
It just seems strange to me how trigonometry is so old although there were no calculating devices back then. I get they might have found traingle sides related to the other sides for 45-45-90 and 30-60-90 triangles using construction or a^2+b^2=c^2 but what about other triangles? Was trigonometry useless except for the 2 previously mentioned triangles before addition and subtraction identities were discovered and others that were necessary for calculating other angles? Was there other ways?

#### Subhotosh Khan

##### Super Moderator
Staff member
It just seems strange to me how trigonometry is so old although there were no calculating devices back then. I get they might have found triangle sides related to the other sides for 45-45-90 and 30-60-90 triangles using construction or a^2+b^2=c^2 but what about other triangles? Was trigonometry useless except for the 2 previously mentioned triangles before addition and subtraction identities were discovered and others that were necessary for calculating other angles? Was there other ways?
They had a "thing" called trig-table (my grandfather did!) and lots of patience.

There are series expansion available for all the trig. functions. It can calculate all the values (to any accuracy) - Just not as fast........

They calculated the height of Mt. Everest - well before anybody set foot on it. Surveying is all trigonometry.

Remember we visited the Moon without a calculator!!

#### Dr.Peterson

##### Elite Member
I used trig tables in high school; that's not long ago, considering ...

When you say trig is old, are you aware how old it is? Trig tables were made at least as far back as Hipparchus (before 125 BC); this Wikipedia article mentions some of the techniques used, largely trig identities, which they already knew (though in slightly different forms). A couple hundred years later, Ptolemy made a famous table that is explained in this article. It's been suggested that the existence of such tables (which took a lot of work) is the reason we still use degrees, which were a standard unit back then -- changing would have required recreating all those tables!

#### MegaMoh

##### New member
They had a "thing" called trig-table (my grandfather did!) and lots of patience.

There are series expansion available for all the trig. functions. It can calculate all the values (to any accuracy) - Just not as fast........

They calculated the height of Mt. Everest - well before anybody set foot on it. Surveying is all trigonometry.

Remember we visited the Moon without a calculator!!
I used trig tables in high school; that's not long ago, considering ...

When you say trig is old, are you aware how old it is? Trig tables were made at least as far back as Hipparchus (before 125 BC); this Wikipedia article mentions some of the techniques used, largely trig identities, which they already knew (though in slightly different forms). A couple hundred years later, Ptolemy made a famous table that is explained in this article. It's been suggested that the existence of such tables (which took a lot of work) is the reason we still use degrees, which were a standard unit back then -- changing would have required recreating all those tables!
of course, they had tables not so long ago. I am talking about when trigonometry was first "founded". They set "sine" to opposite/hypotenuse, "cosine" to adjacent/hypotenuse and "tangent" to opposite/hypotenuse, then what? how did they first find out what sine, say, 15 is? or tangent of 81. They found it somehow which then helped them in making these "tables". My question is what is that "somehow" and how did they find it?

#### Subhotosh Khan

##### Super Moderator
Staff member
of course, they had tables not so long ago. I am talking about when trigonometry was first "founded". They set "sine" to opposite/hypotenuse, "cosine" to adjacent/hypotenuse and "tangent" to opposite/hypotenuse, then what? how did they first find out what sine, say, 15 is? or tangent of 81. They found it somehow which then helped them in making these "tables". My question is what is that "somehow" and how did they find it?
".... what is that "somehow" and how did they find it? "
I do not know for sure.I suspect - like I said in my previous post - sometimes through direct measurement, sometimes through "double angle & half angle formulae and sometimes through series expansion.

#### Dr.Peterson

##### Elite Member
of course, they had tables not so long ago. I am talking about when trigonometry was first "founded". They set "sine" to opposite/hypotenuse, "cosine" to adjacent/hypotenuse and "tangent" to opposite/hypotenuse, then what? how did they first find out what sine, say, 15 is? or tangent of 81. They found it somehow which then helped them in making these "tables". My question is what is that "somehow" and how did they find it?
Did you read the Wikipedia articles I gave you? They answer your questions! I told you about "when trigonometry was first founded", and that they used identities. Surely you know how to find sin(15°), namely by finding that sin(30°) = 1/2 and using the half-angle identity. That's the sort of thing they did (though apparently they started with 72° rather than 60°).

As I mentioned, things were in slightly different forms, as they didn't define exactly the same trig functions we have now; also, I can't give you complete instructions for doing exactly what they did to make the entire table. But they used a sequence of identities (sum, difference, and half-angle) to obtain functions of small angles, and then to put those together to get larger ones.

The article on Ptolemy's table gives a lot of detail, though it will look foreign to you because he was calculating a different function ("chord θ") and using a different number system (base 60), but it's all there to read.

There are also references to subsequent tables that you can look up, to see how things changed.

#### MegaMoh

##### New member
Did you read the Wikipedia articles I gave you? They answer your questions! I told you about "when trigonometry was first founded", and that they used identities. Surely you know how to find sin(15°), namely by finding that sin(30°) = 1/2 and using the half-angle identity. That's the sort of thing they did (though apparently they started with 72° rather than 60°).

As I mentioned, things were in slightly different forms, as they didn't define exactly the same trig functions we have now; also, I can't give you complete instructions for doing exactly what they did to make the entire table. But they used a sequence of identities (sum, difference, and half-angle) to obtain functions of small angles, and then to put those together to get larger ones.

The article on Ptolemy's table gives a lot of detail, though it will look foreign to you because he was calculating a different function ("chord θ") and using a different number system (base 60), but it's all there to read.

There are also references to subsequent tables that you can look up, to see how things changed.
so according to what you just said, trigonometric functions were useless before discovering the half, double, and addition identities? that's what got me baffled in the first place, how would they be useful without knowing angles other than 0-30-45-60-90 and maybe 36 and 72. You said it might be by measurement but would they really rely on measurement for a good amount of time before finding the identities?

#### Dr.Peterson

##### Elite Member
so according to what you just said, trigonometric functions were useless before discovering the half, double, and addition identities? that's what got me baffled in the first place, how would they be useful without knowing angles other than 0-30-45-60-90 and maybe 36 and 72. You said it might be by measurement but would they really rely on measurement for a good amount of time before finding the identities?
No, I don't think I said anything about measuring anything.

Essentially, what I said is that, whatever development the concepts went through initially, the necessary identities were known by the time anyone did anything elaborate with them, because the latter required the tables. We might say (sort of reversing what you just said) that the discovery of various identities was what made the trig functions useful, so either the identities were discovered rather quickly (after all, they knew lots of geometry), or it was the identities that led to using trig functions. I'm not sure that any trig functions were even thought of as things worth naming until identities were known and perhaps until tables were created. (Keep in mind that the very concept of "function" is a relatively recent idea (1800's); a table may have been the closest concept they had. And they had none of the algebraic symbolism we have now.)

Perhaps you have an inaccurate picture of the process of developing a field of mathematics. The sine wasn't lying around somewhere with people wondering, "How can we use this thing?" Rather, people gradually developed theorems about ratios of sides of triangles, discovered identities that could be used to calculate them, thought of problems that could be solved if only they had tables of values, and then made the tables. Only then was there a subject of trigonometry that could be studied. (And the sine, itself, was not given a name until the 500's AD.)

But that is still a vast oversimplification. If you've read the pages I referred you to, you might want to move on to broader stories of the history of trigonometry. Read the whole Wikipedia article, and maybe others such as this and this. It's also worth considering that some of the geometry that Euclid wrote about is equivalent to bits of trigonometry, but without identifying the functions. For example, this theorem is really the Law of Cosines.

#### MegaMoh

##### New member
I see, thank you. I just thought it wouldn't make sense for defining functions to be just a ratio of the side opposite an angle over the longest side in a right triangle while being unable to use them, which is what confused me but now it makes sense that they might have actually found them quickly. But I still don't think they actually had them and thought "if only we could use those with other angles" to be honest. I thought that it was clear how it originated. Thanks!

#### jonah2.0

##### Junior Member
Beer soaked comments follow.
I used trig tables in high school; that's not long ago, considering ...
I remember those days too.
Good times indeed.