The bases of an isosceles trapezoid are 4 and 52. The sides are 25. Find the sine of the acute angle of the trapezoid.

Since the sides are equal in an isosceles trapezoid, their projections onto the larger base are also equal to each other. The length of this projection can be found as half the difference between the lengths of the bases: (52 – 4) / 2 = 48/2 = 24.

The acute angle of this trapezoid is the angle between the large base and the lateral side. The ratio of the length of the projection of the lateral side to the length of the lateral side is the cosine of the acute angle of the trapezoid: cos α = 24/25.

Knowing that cos2α + sin2α = 1, we find the sine of the acute angle of the trapezoid:

sin2α = 1 – cos2α = 1 – (24/25) ^ 2 = 1 – 576/625 = (625 – 576) / 625 = 49/625;

sin α = √ (49/625) = 7/25.




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