IIT JEE , AIEEE Forum - Mathematics , Trignometry

Ricky_rock

If a, b, c and k are constant quantities

and alpha , beta and gama are variable

subjects to the relation atan alpha + b

tan beta + c tan gama=k, find the

minimum value of tan square alpha+tan

square beta + tan square gama.

ramkumar_november

here is my proof:::

we use

the Cauchy?Schwarz inequality is

$\left(\sum_{i=1}^n x_i y_i\right)^2\leq \left(\sum_{i=1}^n x_i^2\right) \left(\sum_{i=1}^n y_i^2\right).$

or simply by putting n=3 we get

$(x_1y_1+x_2y_2+x_3y_3)^2\leq (x_1^2+x_2^2+x_3^2)(y_1^2+y_2^2+y_3^2)$

$put\;\;x_1=a\;\;\;x_2=b\;\;\;x_3=c$

$y_1=tan\alpha\;\;\;y_2=tan\beta\;\;\;y_3=tan\gamma$

we get

$(atan\alpha+btan\beta+ctan\gamma)^2\leq(a^2+b^2+c^2)(tan^2\alpha+tan^2\beta+tan^2\gamma)$

$k^2\leq(a^2+b^2+c^2)(tan^2\alpha+tan^2\beta+tan^2\gamma)$

$tan^2\alpha+tan^2\beta+tan^2\gamma\geq\frac{k^2}{a^2+b^2+c^2}$

so minimum value is $\frac{k^2}{a^2+b^2+c^2}$