In triangle ABC, angle C is 90 degrees, cosA = 0.4, bc = 3√21. Find AB.

From the graphical representation of the problem, we find that AB is the hypotenuse, AC and BC are legs.
Using the formula for the cosine of a right-angled triangle, we get:
cos A = b / c = AC / AB.
But we will not be able to use the formula, since the AU side is not known. Instead of the formula for the cosine of an angle, we will use the formula for the sine of the angle of a right-angled triangle:
sin A = a / c = BC / AB.
Therefore, it is necessary to determine sin A. To do this, we use the main trigonometric property:
sin ^ 2 α + cos ^ 2 β = 1.
Hence:
sin ^ 2 A = 1 – cos ^ 2 A.
sin A = √ (1 – cos ^ 2 A).
sin A = √ (1 – 0.4 ^ 2) = √ (1 – 0.16) = √0.84.
Let’s calculate the side AB.
sin A = BC / AB.
AB = BC / sin A = 3√21 / √0.84 = √ (3 ^ 2 * 21) / √0.84 = √ (9 * 21) / √0.84 = √189 / √0.84 = √ (189 / 0.84) = √225 = 15.

Answer: side AB = 15.