Explain what the Law of sines becomes when one of the angles is a right angle?

Answer 1

The Law of sines: a/sin A = b/sin B = c/sin C

If the angle C in the triangle ABC is 90 degrees, then the triangle ABC is a right triangle, where c is the measure of the hypotenuse, a is the measure of the leg opposite the angle A, and b is the measure of the leg opposite the angle B.

Let us observe what happens when sin C = sin 90 degrees = 1.

c/sin C = a/sin A cross multiply;

c sin A = a sin C divide by c both sides;

(c sin A)/c = (a sin C)/c simplify c on the left hand side;

sin A = (a sin C)/c = [(a)(1)]/c = a/c

sin A = (measure of leg opposite the angle A)/(measure of hypotenuse)

From the Law of Cosine we know that cos A= (b^2 + c^2 - a^2)/(2bc). If we substitute a^2 + b^2 for c^2, we have:

cos A = (b^2 + (a^2+ b^2) - a^2 )/(2ab)

cos A = 2b^2 /2ab simplify;

cos A = b/c = (measure of leg adjacent the angle A)/(measure of hypotenuse) Therefore tan A = sin A/cos A =(a/c)/(b/c) = (a/c)(c/b) = a/b = (measure of leg opposite the angle A)/(measure of leg adjacent to angle A). And cot A = cos A/sin A = (b/c)/(a/c) = (b/c)(c/a) = b/a = (measure of leg adjacent to angle A)/(measure of leg opposite the angle A).