Find the length of the longer diagonal of a 
parallelogram two of whose sides are 34mm 
and 94mm; their included angle is 105 degrees

Draw the parallelogram:

  

Draw in the longer diagonal, call its length x:



Erase the upper triangle:



The law of cosines when given two sides and the angle
included between them is:

(Unknown side)² =

(One given side)²+(Other given side)²-2(One given side)(Other given side)(Cosine of given angle)


So 

x² = 34² + 94² - 2(34)(94)cos(105°)

x² = 1156 + 8836 - 6392(-.2588190451)

x² = 11646.37134
     ___________
x = Ö11646.37134

x = 107.918355mm

You should round that to the correct
degree of precision.  The angle given
to the nearest degree corresponds to
the sides given to the nearest two
significant digits.  That would be

x = 110mm

However, when the first two digits are "10",
some mathematicans claim that you should
increase the precision by one significant
digit.  In that case your final answer
would be

x = 108mm

Edwin