Find angle B in the trigonometry problem

Thread starter chwala
Start date Aug 11, 2021
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Angle Trigonometry

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The discussion centers on finding angle B in a trigonometry problem, with various methods proposed for solving it. One approach involves using the Law of Sines and trigonometric identities to establish that angle B equals 30 degrees. Participants explore geometric constructions and algebraic manipulations, emphasizing the need for accurate relationships between triangle sides and angles. The conversation highlights the complexity of the problem and the cleverness of certain solutions, particularly one that proves the similarity of triangles involved. Ultimately, the consensus is that angle B can indeed be determined to be 30 degrees through careful analysis and proof.

Feb 10, 2023

neilparker62

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Charles Link said:

Very clever @neilparker62 :)

To show the triangles are similar, we need to show ##de/da=da/db ##. (These are corresponding sides around the same angle).
Now ## da/db=da/ac=\sin(40)/\sin(80) ##.
Meanwhile ## de/da=\sin(30)/\sin(50) ##.
We need to show ## \sin(40)/\sin(80)=\sin(30)/\sin(50)##.

With ## \sin(80)=2 \sin(40) \cos(40) ##, and ## \sin(30)=1/2 ##, and ## \cos(40)=\sin(50) ##, we have proven the triangles to be similar, and thereby angle ##B=30 ## degrees.

Yes I remembered now - my workings were with sin(100) rather than sin(80) but same thing obviously.