# Understand the problem

[/et_pb_text][et_pb_text _builder_version="3.22.4" text_font="Raleway||||||||" background_color="#f4f4f4" box_shadow_style="preset2" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px"]Prove that a triangle $ABC$ is right-angled if and only if
$$\sin A + \sin B + \sin C = \cos A + \cos B + \cos C + 1$$

[/et_pb_text][et_pb_tabs active_tab_background_color="#0c71c3" inactive_tab_background_color="#000000" _builder_version="3.23.3" tab_text_color="#ffffff" tab_font="||||||||" background_color="#ffffff"][et_pb_tab title="Hint 0" _builder_version="3.22.4"]Do you really need a hint? Try it first!

[/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="3.23.3"]Familiarity with the trigonometric identities associated with a triangle is a must for any aspiring Olympian. Check the list given in the reference.[/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="3.23.3"]$ABC$ is right-angled iff $\cos A\cos B\cos C=0$.[/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="3.23.3"]Show that $\cos A\cos B\cos C=\frac{s^2-(2R+r)^2}{4R^2}$. [/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="3.23.3"]

Combining hints 2 and 3, we see that $ABC$ is right-angled iff $s=2R+r$.

We know that $\sin A+\sin B+\sin C=\frac{s}{R}$ and $\cos A+\cos B+\cos C=1+\frac{r}{R}$,

hence $ABC$ is right-angled iff $\sin A+\sin B+\sin C=1+\cos A+\cos B+\cos C$.

# Similar Problems

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### One comment on “An alluring trigonometric relation and its Implication”

1. DEB JYOTI MITRA says:

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?

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