In triangle ABC, angle C is 90 degrees, CH height AC = 5, AH = 3. Find cosine B.

Since CH is the height of the triangle ABC, the triangles BCH and ACH are rectangular.

In a right-angled triangle ACH, according to the Pythagorean theorem, we determine the length of the leg CH.

CH ^ 2 = AC ^ 2 – AH ^ 2 = 25 – 9 = 16.

CH = 4 cm.

In a right-angled triangle, the sine of its acute angle is the ratio of the length of the opposite leg to the length of the hypotenuse.

Then, in a right-angled triangle ACH:

SinСAH = CH / AC = 4/5 = 0.8.

In a right-angled triangle, the cosine of one acute angle is equal to the sine of another acute angle, then CosABC = SinCAH = 0.8.

Answer: The cosine of the angle ABC is 0.8.