How would you calculate the sin,cos and tan of an angle without using a calculator?

 

Heh, there is always old school (which I've done). Tables, lots and logs of tables, with interpolation.

Believe it or not there was math before calculators or computers. It is probably no coincidence that early computers were used to... create more tables for artillery in the army.

 

For angles less than about 13 degrees, the sine in radians is a very good approximation; in other words sin (x) = x. This substitution is used extensively in physics to get rid of the sin x to make the math easier. Then from the sine you can calculate the cosine and tangent as well (as long as the angle is small).

 

Get a good trigonometry math book and there will be all you ever wanted to know. Some 'old school' guy is probably throwing one out right now.

 

The OP asked about calculating the values, not looking them up in a table or a book. One way they can be approximated is based on Taylor series. See http://en.wikipedia.org/wiki/Trigonometric_functions#Series_definitions
See also the section on computation: http://en.wikipedia.org/wiki/Trigonometric_functions#Computation

 

The approximation appears to be within every 15 degrees. Through trial and error I will see if there is a more accurate approach

 

The OP asked about calculating the values, not looking them up in a table or a book.

Yes, you could resort to this

One way they can be approximated is based on Taylor series.

but he is asking

without using a calculator?

Hiring Rain Man to run mental calculations?

 

Heh, there is always old school (which I've done). Tables, lots and logs of tables, with interpolation.

Believe it or not there was math before calculators or computers. It is probably no coincidence that early computers were used to... create more tables for artillery in the army.

Interpolation seems quite accurate. I try

 

Yes, you could resort to this



but he is asking



Hiring Rain Man to run mental calculations?

Mental calculations are good because it does us good. Using calc/comp technology is not good, there a luxury. Nothing requires a calculator. Faraday, Tesla, Eastwing etc, did they need one? NO

 

Depending on the accuracy you need, a truncated Taylor or Maclaurin series can be quick and easy. It's really just arithmetic.

 

The OP asked about calculating the values, not looking them up in a table or a book. One way they can be approximated is based on Taylor series. See http://en.wikipedia.org/wiki/Trigonometric_functions#Series_definitions
See also the section on computation: http://en.wikipedia.org/wiki/Trigonometric_functions#Computation

Last I checked interpolation is math, using tables. Tables were the way it was done until the late 1970's, they were a part of the math (that or the slide rule). I am of the age that bridges that gap. It was pretty much the way it was done for many centuries previous. The only people deriving sine function numbers were the table makers, every one else used the tables.

It reminds me of a true story about a mathematician whose life's work was to derive Pi to several thousand place. After his death it was discovered he made a math error several hundred places into the problem. Computers weren't even a concept (except for those who knew Babbage) at that time.

 

Depending on the accuracy you need, a truncated Taylor or Maclaurin series can be quick and easy. It's really just arithmetic.

You can do far better than that, see for example:
http://math.sfu.ca/~cbm/aands/page_76.htm

These are not Taylor series, but more like an economised Chebycheff series.

 

Good point and good find.

 

The sin and cos function are 'ratios' of the length of the sides of a right triangle.

Long division is cumbersome, but it will give you accurate answers to as many decimal places precision as you care to work it out.

(I'm old enough to remember school before calculators were made 'affordable' AND portable enough for students to carry to class, I tried to forget the long division, but I couldn't)

 

Good point, but you do not always have a right triangle to work with. It could be a phase angle, a moment in rotation, or the side of the triangle may not be determinable.

I still think a truncated Maclaurin series is the best way to go, especially for trig functions.

 

Sin and cosine function ARE based on the ratio's of the sides of a right triangle.

How they are used in other math problems(even phase angle calculation) does not change that basic fact.

 

I still think a truncated Maclaurin series is the best way to go, especially for trig functions.

Maclaurin (and Taylor) series are generally inefficient for numerical approximation as they give very low error around the expansion point and very high errors at the range limits.

The series in Abramowitz and Stegun that I linked to were not truncated Maclaurin series. For example, if you look at the approximation there for sin(x) using only three terms (4.3.96), the maximum error over the range is a tenth of the error for the truncated Maclaurin series at x=pi/2 (Maclauren series with the same number of terms).

If you are interested, one way better approximations can be generated is by truncated Tchebycheff expansions, these are not optimal but they can be pretty good.

 

@ Kermit2

I'm not disputing that. Just saying that you may not have this information.


Okay, show me the long division calculation for the cosine of an angle of √2∏e radians, where e is the base of the natural log.

 

Sin and cosine function ARE based on the ratio's of the sides of a right triangle.

How they are used in other math problems(even phase angle calculation) does not change that basic fact.

In maths there are many equivalent definitions for sine, in analysis it is usually defined by:

\(sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots\)

It takes a bit of work to show that this is equivalent to the triangle definition but a good first year university analysis course should cover it.

 

Maclaurin (and Taylor) series are generally inefficient for numerical approximation as they give very low error around the expansion point and very high errors at the range limits.

Agreed. I was under the impression that the OP wanted to calculate on specific value. Still, a 4 term Maclaurin series will give you decent results from -∏ to +∏ for sine or cosine. all depends on your accuracy requirements.

The series in Abramowitz and Stegun that I linked to were not truncated Maclaurin series. For example, if you look at the approximation there for sin(x) using only three terms (4.3.96), the maximum error over the range is a tenth of the error for the truncated Maclaurin series at x=pi/2 (Maclauren series with the same number of terms).

But it becomes a matter of effort. The method you linked to is good, but requires the use of tables (a(n) values) and calculations. If you are just interested in a single value, not a viable range, this is more work.

If you are interested, one way better approximations can be generated is by truncated Tchebycheff expansions, these are not optimal but they can be pretty good.

I'll look them up. Thanks.