Question

The sides of a triangle are $\sin \alpha ,\cos \alpha$ and $\sqrt{1+\sin \alpha \cos \alpha }$ for some $0<\alpha <\frac{\pi }{2}$ . Then the greatest angle of the triangle is

• A. $60{}^{\circ }$
• B. $90{}^{\circ }$
• C. $120{}^{\circ }$
• D. $150{}^{\circ }$

Answer 1

Answer: (C)

Let $a=\sin \alpha ,b=\cos \alpha ,c=\sqrt{1+\sin \alpha \cos \alpha }$ then $\cos C=\frac{a^2+b^2-c^2}{2ab}$ $\Rightarrow$ $\cos C=\frac{{\sin }^2\alpha +{\cos }^2\alpha +1-\sin \alpha \cos \alpha }{2\sin \alpha \cos \alpha }$ $\Rightarrow$ $\cos C=-\frac{\sin \alpha \cos \alpha }{2\sin \alpha \cos \alpha }$ $\Rightarrow$ $\cos C=-\frac{1}{2}=\cos 120{}^{\circ }$ $\Rightarrow$ $\angle C=120{}^{\circ }$