In a right-angled triangle ABC, angle A = 90 degrees, AB = 85 cm, height AD is 77 cm Find AC and cos of angle C.

Since AD is the height of triangle ABC, then triangle ABD is rectangular.

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

SinB = AD / AB = 77/85.

In a right-angled triangle, the sine of one acute angle is equal to the cosine of the other angle, then CosC = SinB = 77/85.

In a right-angled triangle ACD Sin ^ 2C = 1 – Cos ^ 2C = 1 – 5929/7225 = 1296/7225.

SinC = 36/85.

Then SinC = BP / AC.

AC = AD / SinC = 77 / (36/85) = 77 * 85/36 = 181.8 cm.

Answer: CosC = 77/85, AC = 181.8 cm.