In a parallelogram, the sides are 8 and 12 and the acute angle is 60 degrees. Find the diagonals of the parallelogram.

Find the smaller diagonal. For calculations, we use the cosine theorem:
(d1) ² = a² + b² – 2 * a * b * cos 60 ° = 8² + 12² – 2 * 8 * 12 * 0.5 = 64 + 144 – 96 = 112;
d1 = √112 = 4 * √7 = 10.583.

Let’s calculate the large diagonal.
Find the second corner of the parallelogram:
180 ° – 60 ° = 120 °;
(d2) ² = a² + b² – 2 * a * b * cos 120 ° = 8² + 12² – 2 * 8 * 12 * (- 0.5) = 64 + 144 + 96 = 304;
d2 = √304 = 4 * √19 = 17.436.

Answer: The diagonals of the parallelogram are 4 * √7 and 4 * √19, which is approximately 10.583 and 17.436.