Calculus question

Question

Two sides of a parallelogram meet at an angle of 50°. If the length of one side is 3 meters and the length of the other side is 5 meters, find the length of the longest diagonal and the angles that it forms with each of the given sides. Using Law of Sines to find two angles. Show all steps.

Answer 1

Use the Law of Cosine to get the length of the longer diagonal:

c^2 = 3^2 + 5^2 - 2 (3)(5) Cos 130 = 53.2836 so c = 7.299 almost 7.3.

To get the angle that it makes with the side which is = 5 we use the Law of Sines:

Sin(x) / 3 = sin(130) / 7.3 so we have sin(x) = 0,3148 and x = 18.3 degrees. Obviously the other angle is 180-130- 18.3 = 31.7,

Answer 2

No need for the Law of Sines. Consecutive angles are supplementary.

Use Law of Cosines for length of diagonal.

d² = 3² + 5² - 2(3)(5) cos 130°