Brushing Up Math Knowledge: Basic Trig Questions

[gibberingmouther]

i'm going back to school in September. I'm brushing up on my math knowledge/skills now and thinking of maybe joining a math club when i am back in school. I've often thought if i had an expert to talk to i could learn math a lot quicker than relying on just written instruction ... so maybe i could hone my skills by talking with other math-inclined people at school. anybody else think this way?

anyway ... i know sin and cos are are calculated manually and put into tables which you can get from a scientific or graphing calculator. and tanx is just sinx/cosx. are the inverse trig functions calculated on the calculator using those values as well or were the values just added into a table? i know given say cos(5/8) = .625, we know arccos(.625) = 5/8. iff cos(x) = theta, arccos(theta) = x, right? but does anyone know, do the calculators just use table values for inverse trig functions or is there another way of getting the values from the regular sin and cos trig tables?

 

[DrClaude]

Trigonometric functions are not simply based on tables. Implementations will use a mixture of algorithms in order to achieve efficiency and accuracy for any argument. So in addition to table lookup, you will also find, for instance, Taylor series. You can check out for instance how sin() in implemented in gcc:
https://sourceware.org/git/?p=glibc.git;a=blob;f=sysdeps/ieee754/dbl-64/s_sin.c

Inverse trigonometric follow a similar approach.

 

[phinds]

anyway ... i know sin and cos are are calculated manually and put into tables which you can get from a scientific or graphing calculator.

Now stop and think about that for a minute. Suppose you need the sin of 23.4572 degrees. Do you think there's a table that contains that value? And then next it contains the value of sin of 23.4573 degrees? Now, we've left out all the values BETWEEN 23.4572 and 23.4573 but forgetting that for a minute, just think of the size of storage that you would need to contain the values for every sin of x where x is any 6-digit number from 0 to 360 degrees. THEN you need a similar table for each of the other functions you want to use. Now, do you really think they are all contained in tables?

 

[gibberingmouther]

Now stop and think about that for a minute. Suppose you need the sin of 23.4572 degrees. Do you think there's a table that contains that value? And then next it contains the value of sin of 23.4573 degrees? Now, we've left out all the values BETWEEN 23.4572 and 23.4573 but forgetting that for a minute, just think of the size of storage that you would need to contain the values for every sin of x where x is any 6-digit number from 0 to 360 degrees. THEN you need a similar table for each of the other functions you want to use. Now, do you really think they are all contained in tables?

i guess i just assumed they interpolated between the table values.

 

[phinds]

i guess i just assumed they interpolated between the table values.

You're missing my point, which is the SIZE of the table required.

 

[CWatters]

Apparently many calculators use CORDIC...

https://en.m.wikipedia.org/wiki/CORDIC

 

[Mark44]

know given say cos(5/8) = .625, we know arccos(.625) = 5/8.

These are incorrect, but will be correct if you remove "cos" from the first equation and "arccos" from the second. IOW, as 5/8 = .625 -- the fraction 5/8 in decimal form is .625.

 

[gibberingmouther]

These are incorrect, but will be correct if you remove "cos" from the first equation and "arccos" from the second. IOW, as 5/8 = .625 -- the fraction 5/8 in decimal form is .625.

cos(5/8) = .811 so arccos(.811) = 5/8 then, right? i used a random internet calculator because i can't find my graphing calculator. i know it's somewhere...

 

[symbolipoint]

Textbooks should contain trigonometric tables. Handbooks are or were made which contained more accurate trigonometric tables. Any public library should have a Handbook Of Chemistry & Physics and these will have those tables. A scientific calculator or graphing calculator is more convenient.

A brief Google search is unsatisfying. I spent a few minutes (maybe 10 or less) and the only good sites found are these:
http://www.math.com/tables/trig/tables.htm
http://www.mathwarehouse.com/trigonometry/sine-cosine-tangent-chart.php

 

[256bits]

There's the machinist's handbook.
http://www.nashua.edu/paradisem1/Machinery's%20Handbook%2027th/27_math_01.pdf
Tables to minutes of a degree on page 99.
Series expansion on page 103.
Lot's of other good tidbits.

 

[mfb]

i know sin and cos are are calculated manually and put into tables which you can get from a scientific or graphing calculator.

What do you mean by "manually"? People computed them with pen and paper before computers existed, but since then computers are faster (by orders of magnitude). Every calculation you can do with pen and paper can also be done by a computer - and much faster.

You can calculate them without any tables as basis if you calculate enough terms of Taylor expansions and similar things. That is time-consuming, but it gives you a table that can be used for calculations later which can use simpler and faster interpolation methods.

 

[DrClaude]

You're missing my point, which is the SIZE of the table required.

But actual implementations do use tables with interpolation.

 

[phinds]

But actual implementations do use tables with interpolation.

Really? I was not aware of that. Thanks.

 

[Mark44]

But actual implementations do use tables with interpolation.

Really? I was not aware of that. Thanks.

Do a search for CORDIC... I'm told that a lot of calculators use this method for generating values of trig functions.

 

[DrClaude]

Do a search for CORDIC... I'm told that a lot of calculators use this method for generating values of trig functions.

The Wikipedia page on CORDIC is a good start. I don't know about calculators, but I was specifically referencing compilers.