How can I prove the sine theorem?

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Homework Help Overview

The discussion revolves around proving the sine theorem in the context of triangle ABC, specifically the relationship sin(A)/a = sin(B)/b = sin(C)/c.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to prove the sine theorem by drawing a triangle but expresses difficulty in doing so. Some participants suggest geometric interpretations involving vectors and the circumcircle of the triangle.

Discussion Status

Participants are exploring various geometric approaches to the sine theorem, with some providing insights into vector relationships and properties of inscribed triangles. There is no explicit consensus, but multiple lines of reasoning are being examined.

Contextual Notes

The original poster mentions a lack of familiarity with sharing images, which may limit the visual representation of their attempts. There is also a reference to the cosine theorem, indicating a potential area of confusion or connection.

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Homework Statement



How can I prove this theorem in the triangle ABC? sin(A)/a=sin(B)/b=sin(C)/c



Homework Equations



A*B and ...

The Attempt at a Solution



I have drawn a triangle and tried to prove it, but i couldn't. (I don't know how to send a picture to my post!) I know how to prove the cosine theorem but i can't prove the sine theorem.
 
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Consider the three vectors A to B, B to C and C to A. Now consider geometrically their cross products. The magnitude of each is equal to twice the area of the triangle, so the three are equal.
 
I particularly like this one:

For any triangle, inscribe it in a circle. So the triangle touches the circle at 3 points, call them A, B, and C, and their corresponding angles a, b, and c, and the lengths of the sides opposite them L(a), L(b), L(c).

Move point A along the circumference so that AB passes the center of the circle. Angles subtended by equal chords onto the circumference are equal, so angle BAC is still a.

Now we have a triangle in a semi circle, and so angle ACB is equal to 90 degrees. By trigonometry, sin a = L(a)/Diameter, and so Diameter = sin a/L(a)

Making similar transformations for the other sides shows sin b/L(b) and sin c/L(c) are also equal to the diameter, and hence equal to each other.
 
Oh, thank you very much really.
 

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