If the cosines of the two of the angles of a triangle are proportional to the oppsite sides show that the trangle is right angled.?

2 Answers
Aug 13, 2018

Such triangles are isosceles, inclusive of isosceles right angled triangles.

Explanation:

a / b = sin A/sin B = cos A/cos B, giving

#sin ( A - B ) = sinA cos B - cos A sin B = 0.So.

A = B, and so, A + B + C = 2A + C = pi

rArr the triangle is isosceles that also could be right angled.

Aug 13, 2018

If we consider that the cosines of the two of the angles of a triangle are inversely proportional to the opposite sides then we can get the expected result. Let for DeltaABC

acosA=bcosB

=>2RsinAcosA=2RsinBcosB

=>sin2A=sin2B

=>sin2A=sin(pi-2A)

=>2A=pi-2A

=>A+B=pi/2

Hence C=pi/2. This means the triangle is right angled when cosine of two angles are inversely proportional to opposite sides.