In triangle ABC, altitudes AD and BE are drawn to the corresponding bases. If ∠BAC = 45° and ∠ABC = θ, then \(\rm \frac{AD}{BE}\) equals

Here we know, ∠BAC = 45° and ∠ABC = θ

We know when 2 sides are given and the angle between them is given:

Area = 1/2 × side1 × side2 × Sin(angle)

Area of ΔABC = 1/2 × AB × BC × Sinθ = 1/2 × AB × AC × Sin45

=> AB × BC × Sinθ /2 = AB × AC / 2√2   => AC / BC = √2 Sinθ

Now as AD and BC are altitudes :

The area of triangle can also be written as 1/2 × AD × BC = 1/2 × AC × BE

AD / BE = AC/BC which happens to be √2 Sinθ

Hence AD/BE = √2 Sinθ