prove that if in triangle ABC tan(A/2)=5/6 tan(B/2)=2/5 then a,b,c will be in AP

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- Dec 28th 2010, 08:39 AMprasumtrigo help
prove that if in triangle ABC tan(A/2)=5/6 tan(B/2)=2/5 then a,b,c will be in AP

- Dec 28th 2010, 09:04 AMBAdhi
what's "AP"

- Dec 28th 2010, 09:04 AMPlato
- Dec 29th 2010, 03:44 AMprasum
i have found tan A and Tan B through tan2A formula after that i have found out Tan C by Tan A+TanB+TanC=TanATanBTanC brcause A+b+c=180 degrees after that how should i find a,b,c and prove they are in arithmetic progression

- Dec 29th 2010, 08:13 AMBAdhi
wouldn't $\displaystyle sinA=\frac{2tan(A/2)}{1+tan^2(A/2)}$ and $\displaystyle \frac{a}{sin(A)}=\frac{b}{sin(B)}=\frac{c}{sin(C)}$ work?

- Dec 30th 2010, 06:18 AMArchie Meade
$\displaystyle \displaystyle\ tan\left[\frac{A}{2}\right]=\frac{5}{6}\Rightarrow\ sinA=\frac{2tan\left[\frac{A}{2}\right]}{1+tan^2\left[\frac{A}{2}\right]}=\frac{\left[\frac{5}{3}\right]}{\left[\frac{61}{36}\right]}=\frac{60}{61}$

$\displaystyle \displaystyle\ tan\left[\frac{B}{2}\right]=\frac{2}{5}\Rightarrow\ sinB=\frac{\left[\frac{4}{5}\right]}{\left[\frac{29}{25}\right]}=\frac{20}{29}$

$\displaystyle sinC=sin[180^o-(A+B)]=sin(A+B)=sinAcosB+cosAsinB$

We obtain the cosines from Pythagoras' Theorem.

$\displaystyle x^2+60^2=61^2\Rightarrow\ x^2=3721-3600=121\Rightarrow\ x=11$

$\displaystyle \displaystyle\Rightarrow\ cosA=\frac{11}{61}$

$\displaystyle y^2+20^2=29^2\Rightarrow\ y^2=29^2-20^2=441\Rightarrow\ y=21$

$\displaystyle \displaystyle\Rightarrow\ cosB=\frac{21}{29}$

$\displaystyle \displaystyle\ sinC=\frac{60}{61}\;\frac{21}{29}+\frac{11}{61}\;\ frac{20}{29}=\frac{1260+220}{1769}=\frac{1480}{176 9}$

From the Law of Sines

$\displaystyle \displaystyle\frac{a}{sinA}=\frac{b}{sinB}=\frac{c }{sinC}$

$\displaystyle \Rightarrow\ a:b:c=sinA:sinB:sinC$

If a, b, and c are in Arithmetic progression,

then the sum of the 3 sides is 3 times the "middle side".

$\displaystyle p+(p+d)+(p+2d)=3p+3d=3(p+d)$

Hence, the sum of the 3 sines must be 3 times the sum of the "middle sine".

$\displaystyle sinA+sinB+sinC=2.50989259469$

$\displaystyle 3sinC=2.50989259469$

Therefore

a, c, b are in Arithmetic Progression.

b, c, a are in Arithmetic Progression.