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June 13, 2019

An alluring trigonometric relation and its Implication

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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_column][/et_pb_row][et_pb_row _builder_version="3.22.4"][et_pb_column type="4_4" _builder_version="3.22.4"][et_pb_accordion open_toggle_text_color="#0c71c3" _builder_version="3.23.3" toggle_font="||||||||" body_font="Raleway||||||||" text_orientation="center" custom_margin="10px||10px"][et_pb_accordion_item title="Source of the problem" open="on" _builder_version="3.23.3" title_text_shadow_horizontal_length="0em" title_text_shadow_vertical_length="0em" title_text_shadow_blur_strength="0em" closed_title_text_shadow_horizontal_length="0em" closed_title_text_shadow_vertical_length="0em" closed_title_text_shadow_blur_strength="0em"]Vietnam National Mathematical Olympiad 1981[/et_pb_accordion_item][et_pb_accordion_item title="Topic" open="off" _builder_version="3.23.3" title_text_shadow_horizontal_length="0em" title_text_shadow_vertical_length="0em" title_text_shadow_blur_strength="0em" closed_title_text_shadow_horizontal_length="0em" closed_title_text_shadow_vertical_length="0em" closed_title_text_shadow_blur_strength="0em"]Trigonometry[/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" open="off" _builder_version="3.23.3" title_text_shadow_horizontal_length="0em" title_text_shadow_vertical_length="0em" title_text_shadow_blur_strength="0em" closed_title_text_shadow_horizontal_length="0em" closed_title_text_shadow_vertical_length="0em" closed_title_text_shadow_blur_strength="0em"]Medium[/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" open="off" _builder_version="3.23.3" title_text_shadow_horizontal_length="0em" title_text_shadow_vertical_length="0em" title_text_shadow_blur_strength="0em" closed_title_text_shadow_horizontal_length="0em" closed_title_text_shadow_vertical_length="0em" closed_title_text_shadow_blur_strength="0em"]Challenge and Thrill of Pre-college Mathematics[/et_pb_accordion_item][/et_pb_accordion][et_pb_text _builder_version="3.22.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px"]

Start with hints

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[/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.

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Watch the video (Coming Soon)

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Similar Problems

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

  1. 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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