In triangle ABC, the angle c is 90 degrees, the sine of angle A is 7/27. Find the sine of angle B.

Find sin b in triangle abc, if known:

The angle c is 90 °,
sin A = 7/27.

Decision.

sin a = BC / AB (the ratio of the opposite leg to the angle a to the ratio of the hypotenuse);

cos a = AC / AB (the ratio of the adjacent leg to the angle a to the ratio of the hypotenuse);

sin b = AC / AB;

Since, sin b = AC / AB = cos a, then we find sin a by the formula:

cos ^ 2 a + sin ^ 2 a = 1;

cos ^ 2 a + (7/27) ^ 2 = 1;

cos ^ 2 a + 49/729 = 1;

cos ^ 2 a = 1 – 49/729;

cos ^ 2 a = (729 – 49) / 729;

cos ^ 2 a = 680/729;

cos a = √680 / 27 = 2/27 * √170 = sin b.