An isosceles triangle has two sides of length w that make a 2á-degree angle. Write down two different formulas for the area of this triangle, in terms of w and á (Greek “alpha”). By equating the formulas, discover a relation involving sin 2á, siná, and cosá.

Answer 1

For the whole triangle
Area = (1/2)(w)(w)sin 2a = (1/2)w^2sin 2a

draw in an altitude h, bisecting the angle 2a because we have an isosceles triangle
Area of triangle = (1/2)hwsin a + (1/2)whsina
= whsin a

but cos a = h/w
h = wcos a
then whsin a = w(wcos a)sin a = w^2sin a cos a

finally then
(1/2)w^2 sin 2a = w^2 sin a cos a
sin 2a = 2sin a cos a

nice!