The area of the right triangle is 392√3. One of the acute angles is 30 °. Find the length of the leg lying opposite this corner.

Denote:
a – desired leg
b – second leg
c – hypotenuse
sin30 ° = 1/2 (tabular value)
sin30 ° = a / c = 1/2 (by definition of the sine)
c = 2a
By the Pythagorean theorem:
a2 + b2 = c2
a ^ 2 + b ^ 2 = (2a) ^ 2
b ^ 2 = 3a ^ 2
b = a√3
From the condition: Triangle = ab / 2 = 392√3
a * a√3 / 2 = 392√3
Reduce √3:
a ^ 2 = 392 * 2 = 784
a = 28
Answer: a = 28