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October 5th, 2018, 07:00 AM   #1
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law of sines

Why the author use the law of sines? What is goal to use it?
In the link:
https://www.maa.org/external_archive...n/Polygon.html
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October 5th, 2018, 07:43 AM   #2
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That possibly wasn't intended.
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October 5th, 2018, 07:54 AM   #3
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...

Quote:
Originally Posted by skipjack View Post
That possibly wasn't intended.
Excuse me ?!
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October 5th, 2018, 02:11 PM   #4
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Quote:
Originally Posted by shaharhada View Post
Why the author use the law of sines? What is goal to use it?
In the link:
https://www.maa.org/external_archive...n/Polygon.html
The law of sines gives the length of the side opposite the angle, needed to compute the area of the triangle whose other two sides are radii of length one.
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Last edited by skipjack; October 6th, 2018 at 03:24 AM.
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October 6th, 2018, 04:19 AM   #5
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It brings in the sine of another angle, which isn't in the desired area formula.
Area.PNG
By the above diagram, $A(P_nOP_{\small1}) = \frac121\cdot1\cdot\sin(\alpha) = \frac12\sin(\alpha)$.
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October 6th, 2018, 02:10 PM   #6
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Area formula (radius=1). Split triangle in half (right triangles) altitude $=\cos(\frac{\alpha}{2})$, base$=\sin(\frac{\alpha}{2})$
Area of original triangle $=\cos(\frac{\alpha}{2})\sin(\frac{\alpha}{2}) = \frac{1}{2}\sin(\alpha)$.

Last edited by skipjack; October 6th, 2018 at 04:11 PM.
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October 6th, 2018, 05:01 PM   #7
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The half angle identity used above is most easily proved by use of the $\frac12bc\sin(A)$ formula for a triangle's area.
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