# Understand the problem

Prove that a triangle is right-angled if and only if

##### Source of the problem

Vietnam National Mathematical Olympiad 1981

##### Topic

Trigonometry

##### Difficulty Level

Medium

##### Suggested Book

Challenge and Thrill of Pre-college Mathematics

# Start with hints

Do you really need a hint? Try it first!

Familiarity with the trigonometric identities associated with a triangle is a must for any aspiring Olympian. Check the list given in the reference.

is right-angled iff .

Show that .

Combining hints 2 and 3, we see that is right-angled iff .

We know that and ,

hence is right-angled iff .

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Prove that a triangle ABC is right-angled if and only if sin A+sin B+sin C= cos A+cos B+cos C+1;

<C=90;sin A+sin B+sin C=(a+b)/c+1; cos A+cos B+cos C=(a+b)/c+0=(a+b)/c;;

hence sin A+sin B+sin C= cos A+cos B+cos C+1 proved ;

how to show that if sin A+sin B+sin C= cos A+cos B+cos C+1 lead it to ABC as rt angled triangle?