In a triangle with sides 2,3 and 4, the cosine of the angle opposite the smaller side is less than two thirds?

Let the angle opposite the smaller side be equal to a.
Formula for calculating the area of a triangle on two sides and the angle between them:
S = (1/2) ab * sin (a).
Area of a triangle according to Heron’s formula:
p = (2 + 3 + 4) / 2 = 9/2;
S = √ (9/2) (9/2 – 2) (9/2 – 3) (9/2 – 4) = √ ((9/2) (5/2) (3/2) (1 / 2)) = (3/4) * √15;
(3/4) * √15 = (1/2) * 3 * 4 * sin (a);
sin (a) = (3 * √15) / (4 * 6) = √15 / 8;
cos (a) = √ (1 – sin ^ 2 (a)) = √ (1 – 15/64) = √ (49/64) = 7/8;
We compare the value of cos a = 7/8 and 3/2, finding a common denominator for the fractions:
2/3 = 16/24;
7/8 = 21/24.
Answer: cos (a)> 2/3.