How where all the possible tangents, cosines, and sines of angles found?

assuredlonewo

How where all the possible tangents, cosines, and sines of angles found?

micromass

Hi assuredlonewo What do you mean with this question. I understand it in three ways:
- How did they come up with the concept of sine
- how did they find the sine of a specific nice angle, say 45°
- how do they find the sine of all possible angles

Which of these (if any) do you mean?

assuredlonewo

Hi assuredlonewo What do you mean with this question. I understand it in three ways:
- How did they come up with the concept of sine
- how did they find the sine of a specific nice angle, say 45°
- how do they find the sine of all possible angles

Which of these (if any) do you mean?
how do they find the sine of all possible angles

Studiot

Trigonometric functions like sin, cos and tan can be calculated as the sum of a series to any desired accuracy by taking sufficient terms.

This is how trigonometric tables were originally prepared.

For example the sin of any angle is given by

$$\sin (x) = x - \frac{{{x^3}}}{{3*2*1}} + \frac{{{x^5}}}{{5*4*3*2*1}} - \frac{{{x^7}}}{{7*6*5*4*3*2*1}}$$

If you ask a computer or calculator for sin(x) it works the value out this way each time, it does not store tables.

go well

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assuredlonewo

Thanks studiot, I am going to have to dive deeper into this.

micromass

I don't know how they did it in the ancient times. It was probably just drawing a large enough triangle and measuring correctly. But right now, they can do it amazingly accurate. This is thanks though a tool called "Taylor series".

Briefly, mathematicians have shown that

$$\sin(x)=\sum_{n=0}^{+\infty}{\frac{(-1)^nx^{2n+1}}{(2n+1)!}}$$.

Of course, this is an infinite sum, so it's not really useful in practice, however, we can limit the sum to obtain a reasonable approximation to the sine. For example, if we take n to 4, then we get

$$\sin(x)\approx x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\frac{x^9}{9!}$$

This approximation won't be exact, but it'll be a good approximation nonetheless. If we take n even bigger, then we get even better approximations for sines.

There is however, no exact formula to calculate sines (unless in specific examples). But for most applications, we have no need for exact formula's...

Deveno

the earliest trigonometric tables were computed using geometry and interpolation. it is possible to construct a (somewhat horrendous) algebraic expression of sin(1 degree) and work from there. many ancient mathematicians devoted a good portion of their lives to creating such tables, which were used as references for centuries.

the concept of sine was not the first basic concept of trigonometry, but came rather later than the concept of a chord (a straight line segment connecting the ends of a circular arc). later, "half-chords" came to be frequently used, these are what we now call sines. all of these developed in the absence of any way of coordinatizing curves.

it was not uncommon, even into the 20th century, for most people using sines to reference tabulated values. for example, when i was young, i owned a book full of such tables, published by the chemical rubber company (and known popularly as "CRC tables"). these are still published, and are a good reference for anyone using mathematics in any kind of professional capacity (filled with all kinds of neat formulas you might want to remember).

the use of infinite series pre-dates calculus somewhat, although even mathematicians got confused as to which series were convergent, and which weren't. it was probably Liebnitz who first found the infinite series for sin(x), allowing values to be calculated to any desired accuracy.

jtbell

Mentor
it was not uncommon, even into the 20th century, for most people using sines to reference tabulated values.
Personal electronic calculators that could do trig functions became available in about the 1972-1975 period. When I was an undergraduate, my roommate had one of the early Texas Instruments calculators which could not do trig functions. This was about 1973. Shortly after I started graduate school in 1975, I bought my first calculator, a Hewlett-Packard which could do trig functions. Before that, I used a book of tables like the CRC book.

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