I have tried and failed to get this to work:

R=2T COS(A/2) (R is the resultant force on a sheave axle where T is rope tension and A is the included angle)

I can easily solve it with a calculator and get the right answer but CALC keeps returning the wrong answer. It must be that I don’t know the right syntax

How about showing us what your calc syntax looks like, and providing some example data with correct and incorrect answers? I would have thought:

If T value is in cell A1, and the angle A is in cell B1, then the formula for R for example in cell C1 = 2*A1*COS(B1/2) would be what you want.

Are you sure the calculator answer is correct? Angle A measured in degrees or radians? Calculator correctly handles the sequence of operations?

You could try breaking down the formula and calculating each part separately, e.g. do you get the same answer from calculator and calc for just COS(A/2) aka =COS(B1/2)?

Edit:

I just found this (Mathematical Functions - LibreOffice Help):

COS

Returns the cosine of the given angle (in radians).

Syntax

COS(Number)

Returns the (trigonometric) cosine of Number, the angle in radians.

To return the cosine of an angle in degrees, use the RADIANS function.

Examples

=COS(PI()*2) returns 1, the cosine of 2*PI radians.

=COS(RADIANS(60)) returns 0.5, the cosine of 60 degrees.

So maybe what you need is:

=2*A1*COS(RADIANS(B1/2))

2*A1*cos(radians(b1/2) worked. I didn’t know about the radians thing - thanks.

There are four easy test cases to check the formula. a 180 degree included angle gives a value of 0. A 120 degree included angle returns a value equal to rope tension. A 90 degree included angle returns 1.41 times the tension. a 0 degree included angle returns 2x the tension.