Explanation:
- remember, the actual lengths of trigonometric lengths for a given triangle is always the trigonometric function (sine, cosine, tangent,...) result times the length of the actual r (radius) of the circle we need to imagine to be there, that is around the trigonometric triangle.
let's call the missing distance from the foot of the tree to the 60° angle = x.
h = height of the tree.
when we talk about tangent, then r (the radius of the invisible circle around the triangle) is the distance from the tree to the angle (60° and 30°). as the angle is at the center of the invisible circle.
r of the small right-angled 60° triangle is therefore x.
r of the large right-angled triangle with 30° is 50+x.
in both cases the tree has the same height, of course.
so,
h = tan(60)×x = tan(30)×(x + 50) =
= tan(30)×x + tan(30)×50
tan(60) = sqrt(3)
tan(30) = 1/sqrt(3)
h = sqrt(3)×x = x/sqrt(3) + 50/sqrt(3)
sqrt(3)×x = x/sqrt(3) + 50/sqrt(3)
now we multiply both sides by sqrt(3)
sqrt(3)×sqrt(3)×x = x×sqrt(3)/sqrt(3) + 50×sqrt(3)/sqrt(3)
3×x = 3x = x + 50
2x = 50
x = 50/2 = 25 m
and we remember
h = tan(60)×x = sqrt(3)×25 = 25×sqrt(3) m
that is exactly the given height in the problem statement.
so, it is proven that this is correct.