I Thought It Adds Up To 180?

Geometry Level 2

sin ⁡ ∠ A B C = sin ⁡ 13 7 ∘ sin ⁡ ∠ B A C = sin ⁡ 7 6 ∘ sin ⁡ ∠ A C B = sin ⁡ 6 1 ∘ \begin{array} {l l } \sin{\angle ABC}& =\sin{137^{\circ}} \\ \sin{\angle BAC}& =\sin{76^{\circ}} \\ \sin{\angle ACB}& =\sin{61^{\circ}}\ \end{array}

Is it possible to have a △ A B C \triangle ABC satisfying the conditions above?

Yes No

Sam Bealing by Sam Bealing , 21, United Kingdom

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1 solution

Sam Bealing
Jun 12, 2016

Relevant wiki: Triangles Problem Solving - Basic

Note sin ⁡ 13 7 ∘ = sin ⁡ 18 0 ∘ − 13 7 ∘ = sin ⁡ 4 3 ∘ \sin{137^{\circ}}=\sin{180^{\circ}-137^{\circ}}=\sin{43^{\circ}} so let:

∠ A B C = 4 3 ∘ , ∠ B A C = 7 6 ∘ , ∠ A C B = 6 1 ∘ \angle ABC=43^{\circ},\angle BAC=76^{\circ}, \angle ACB=61^{\circ}

4 3 ∘ + 7 6 ∘ + 6 1 ∘ = 18 0 ∘ 43^{\circ}+76^{\circ}+61^{\circ}=180^{\circ}

So it is possible and the answer is:

Yes \color{#20A900}{\boxed{\boxed{\text{Yes}}}}

12 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

Great question!

How can we generalize this problem?

Yes, it exactly follows from the property: sin ⁡ x = sin ⁡ ( π − x ) \sin x=\sin(\pi-x) . BTW (+1)...

Rishabh Jain - 5 years ago

Beautiful question :)

Abhay Tiwari - 5 years ago

Nice question and suitable title.

Ashish Menon - 4 years, 10 months ago

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