Question

In $ \Delta \,\,ABC $ if $ si{{n}^{2}}A+si{{n}^{2}}B+si{{n}^{2}}C=2, $ then the triangle is

• A. right angled, but need not be isosceles
• B. right angled and isosceles
• C. isosceles, but need not be right angled
• D. equilateral

Answer 1

Answer: (A)

Given, $ {{\sin }^{2}}A+{{\sin }^{2}}B+{{\sin }^{2}}C=2 $
$ \Rightarrow $ $ 1-{{\cos }^{2}}A+1-{{\cos }^{2}}B+1-{{\cos }^{2}}C=2 $
$ \Rightarrow $ $ 1={{\cos }^{2}}A+{{\cos }^{2}}B+{{\cos }^{2}}C $
$ \Rightarrow $ $ 1=1-2\cos A\,\cos B\,\cos C $
$ \Rightarrow $ $ \cos A\,\cos B\,\cos \,C=0 $
At least one angle should be $ {{90}^{o}} $ and sum of two angles should be $ {{90}^{o}} $ .
Hence, option [a] is correct.