that law states:
c^2 = a^2 + b^2 - 2ab*cos(C)
C is the angle between sides a and b.
a is one side of the parallelogram.
b is the other side of the parallelogram.
c is the side opposite angle C.
you get:
a = 74
b = 11
C = 45 degrees.
using that formula and the values of a and b, you get:
c^2 = a^2 + b^2 - 2ab*cos(C) becomes:
c^2 = 74^2 + 11^2 - 2*74*11*cos(45) which becomes:
c^2 = 4445.83016.
c = sqrt(c^2) = 66.67705873.
the adjacent angle is equal to 135 degrees.
to find the other diagonal, use the same formula.
in this case:
a = 74
b = 11
C = 135 degrees.
the formula of c^2 = a^2 + b^2 - 2ab*cos(C) which becomes:
c^2 = 74^2 + 11^2 - 2*74*11*cos(135) which becomes:
c^2 = 6748.16984.
c = sqrt(c^2) = 82.14724487.
the lengths of your two diagonals are:
longer diagonal = 82.14724487.
short diagonal = 66.67705873.
those are your solutions.
there is a calculator online that does this for you, so you can confirm that your calculations are correct.
that calculator can be found at http://planetcalc.com/1149/
a picture of that calculator's output is shown below:
a picture of what the parallelogram looks like is shown below:
the drawing is not to scale. the ratio of the length to the width is less than it would be if it were drawn to scale. this was done to showcase the diagonals better because they would have been squeezed together too much to show them clearly otherwise.
it is possible to solve this without using the law of cosines, but all you are doing is proving that the law of cosines will give you the correct answer, and it's a lot more labor intensive than just using the law of cosines.