Ration of sin of angles

Dec 2010
107
3
In a \(\displaystyle \Triangle ABC\;,\) If \(\displaystyle \tan A:\tan B;\ \tan C = 1: 2:3\;,\) Then \(\displaystyle \sin A: \sin B: \sin C,\) is
 

chiro

MHF Helper
Sep 2012
6,608
1,263
Australia
He jacks.

Hint - Tan(x) = sin(x)/cos(x) along with the understanding of a completely positive relationship (which will limit quadrants).
 
May 2009
612
334
In a \(\displaystyle \Triangle ABC\;,\) If \(\displaystyle \tan A:\tan B;\ \tan C = 1: 2:3\;,\) Then \(\displaystyle \sin A: \sin B: \sin C,\) is
To show a triangle in LaTeX, the code is \bigtriangleup, not \triangle:
\(\displaystyle \bigtriangleup ABC\)

Let tan A = k, tan B = 2k, and tan C = 3k. Since we have a triangle ABC, use the identity
\(\displaystyle \tan A + \tan B + \tan C = \tan A \tan B \tan C\),
make substitutions, and solve for k. (There will actually be 3 solutions for k, but only one will be valid.) You'll then find that angle A is a special angle. Angles B and C are not "nice" angles (not whole numbers in degrees), however.


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