In triangle ABC, angle C = 90 degrees, sin A = 0.4; AC = 3√21. Find AB.

It is known that the sine of an angle in a right-angled triangle is equal to the ratio of the opposite leg to the hypotenuse.

By condition, sin A = 0.4 and the adjacent leg AC = 3 (21) ^ 0.5 are given in a right-angled triangle ABC.

You can calculate the cosine of angle A using the trigonometric identity

(cos A) ^ 2 + (sin A) ^ 2 = 1.

Where, (1 – (sin A) ^ 2) ^ 0.5;

cos A = (1 – 0.4 ^ 2) ^ 0.5 = 0.4 (21) ^ 0.5.

On the other hand, cos A = AC / AB and

the hypotenuse is equal to AB = AC / cos A;

AB = 3 (21) ^ 0.5 / 0.4 (21) ^ 0.5 = 7.5.