In triangle ABC, ∠A = 45° and ∠B = 60°. The ratio a : b of the sides opposite them is:

The sine rule gives a : b = sinA : sinB = sin45° : sin60° = (1/√2) : (√3/2). Multiplying both parts by 2√2 clears the fractions: 2 : √6, which is √2 : √3 after dividing by √2. Numerically 0.7071/0.8660 = 0.8165 = √2/√3. Inverting the rule gives √3 : √2 ≈ 1.22, which would make the side facing 45° the longer one — impossible, since a bigger angle always faces a bigger side. Comparing the angles themselves gives 45 : 60 = 3 : 4, and 1 : √3 is the ratio belonging to 30° and 60°.