The Law of Cosines is a beautiful formula and a gateway to all sorts of wonders.
This question asking for the length to the nearest whole number is pretty ugly.
A diagonal of a parallelogram makes two congruent triangles. In this problem we're almost told we're interested in the diagonal opposite an A=55 degree angle, included between sides b=4 and c=6.
[tex]a^2 = b^2 + c^2 - 2 b c \cos A[/tex]
We just plug in the numbers.
[tex]a^2 = 4^2 + 6^2 - 2(4)(6) \cos 55^\circ[/tex]
[tex]a^2 = 52 - 48 \cos 55^\circ[/tex]
That's the exact answer, its square anyway. Now we approximate.
[tex]\cos 55^\circ \approx .57[/tex]
[tex]a^2 = 52 - 48(.57) = 24.6[/tex]
Taking the square root to the nearest integer,
[tex]a = 5[/tex]
