How to Increase the accuracy of trigonometric functions?

The typical function "sin", "cos" or "tan" seems not accurate enough for me. How to improve it?

And the function "digits" didn't work in this case.

Example codes (The answers should be 0):

>> alpha = 1;

>> cos(alpha) * tan(alpha) - sin(alpha)

ans =

1.1102e-16

>> tan(alpha) - sin(alpha) / cos(alpha)

ans =

2.2204e-16

>> digits(256);

>> cos(alpha) * tan(alpha) - sin(alpha)

ans =

1.1102e-16

>> tan(alpha) - sin(alpha) / cos(alpha)

ans =

2.2204e-16

Answer 1

[Walter Roberson]

alpha = sym(1);
>> simplify(cos(alpha) * tan(alpha) - sin(alpha))
 
ans =
 
0
 
>> simplify(tan(alpha) - sin(alpha) / cos(alpha))
 
ans =
 
0

There is no realistic prospect of getting an exact 0 for these calculations when using finite binary floating point calculations. The problem is not the accuracy of the trig functions: the problem occurs for all finite representations http://matlab.wikia.com/wiki/FAQ#Why_is_0.3_-_0.2_-_0.1_.28or_similar.29_not_equal_to_zero.3F

[Steven Lord]

Adding to what Walter said, the digits function has no effect on computations using just numeric variables. It affects symbolic calculations only.

Depending on what you're trying to do the sinpi family of functions (introduced in release R2018b) or the sind family of functions may be of use to you. See the list of functions on this documentation page to know what's available.

[Walter Roberson]

For example in ieee 754 binary double precision

1/49*49-1

is not 0.