Question

angles of a triangle are A,B and C. find the minimum value of \(\large tan^2 \frac{A}{2}+tan^2 \frac{B}{2}+tan^2 \frac{C}{2} \)

Answer 1

hint
\( \large a^2+b^2+c^2 \ge ab+ac+bc \)

Answer 2

hint

\(\huge \frac{B}{2}+\frac{C}{2}=\frac{\pi}{2}-\frac{A}{2} \)

Answer 3

For the special case of an isosceles triangle with B one of the equal angles, the problem expression can be converted to the following expression\[2 \tan ^2\left(\frac{B}{2}\right)+\tan ^2\left(\frac{1}{2} (\pi -2 B)\right) \]The derivative is\[2 \text{Sec}\left[\frac{B}{2}\right]^2 \text{Tan}\left[\frac{B}{2}\right]-2 \text{Sec}\left[\frac{1}{2} (-2 B+\pi )\right]^2 \text{Tan}\left[\frac{1}{2} (-2 B+\pi )\right] \]
The minimum function value is one, at B = pi/3.

Answer 4

\(\large cot(\frac{B}{2}+\frac{C}{2})=cot(\frac{\pi}{2}-\frac{A}{2}) \) then we have \(\large \frac{1-tan(\frac{B}{2})tan(\frac{C}{2})}{tan(\frac{B}{2})+tan(\frac{C}{2})}=tan(\frac{A}{2}) \) ; cross multiply it will give

\( \large tan(\frac{A}{2})tan(\frac{B}{2})+tan(\frac{A}{2})tan(\frac{C}{2})+tan(\frac{B}{2})tan(\frac{C}{2})=1 \)
now use first hint

Answer 5

I think ge will come..

Answer 6

\[\tan^2(\frac{A}{2}) + \tan^2(\frac{B}{2}) + \tan^2(\frac{C}{2}) \ge 1\]

Answer 7

So minimum value will be 1..

Answer 8

and do u know when equality occurs?

Answer 9

I did not get what you asked..

I want to ask how it is greater than or not less than...???

Answer 10

see my first reply mere yaar

Answer 11

That I know..

The hint is saying ab + bc + ca <= a^2 + b^2 + c^2..

but when we converted ab + bc into a^2 + b^2 etc etc then why we put greater sign there and not less than sign??

Answer 12

Yes yes I got it..

Answer 13

if a = b,

and a < c

then b will also less than c that is : b < c

Answer 14

or simply put its values in the hint:

\[\large a^2+b^2+c^2 \ge ab+ac+bc\]

\[\large a^2+b^2+c^2 \ge 1\]

Answer 15

nope
well see this
for any triple of real numbers like a,b,c u have \( (a-b)^2+(a-c)^2+(b-c)^2 \ge 0 \) simplify it u will \( a^2+b^2+c^2 \ge ab+ac+bc \)

Answer 16

let a=tan A/2 b=tan B/2 and c=tan C/2

Answer 17

Yes benim yoldashim I got this one but my doubt was something else and now it has been cleared...

Answer 18

how about the minimum value
when the minimum occurs?

Answer 19

I have told you earlier in my post..

Answer 20

Minimum value will be 1..

Answer 21

i mean for which triangle with a particular property?

Answer 22

Since it is equal or greater than 1 so it will start from 1..

So, 1 is the minimum value..

Answer 23

note that the equality occurs when a=b=c am i right?

Answer 24

Yes..

Answer 25

that gives tan A/2=tan B/2=tan C/2 or A=B=C a ...... triangle

Answer 26

60 = 60 = 60

Equilateral Triangle..

Answer 27

Exactly

Answer 28

When teacher is so good then why not student..

Ha ha ha..