Trigonometry, find the minimum of tan(a).tan(b).tan(c)

[Michael_Light]

Homework Statement

1. Given a,b,c are acute angles and a + b + c =180. find the minimum of tan(a).tan(b).tan(c)

2. Prove that if a+b+c=90, then tan(a)+tan(b)+tan(c) >= 3^1/2

Homework Equations

The Attempt at a Solution

I don't even have any ideas how should i start to find/prove them... any hints?

 

[rock.freak667]

For the first one I'd try to expand the left side after taking the tangent of both sides.

tan(a+b+c) = tan(180°)

 

[ehild]

Following rock.freak's suggestion, I would write the tangent of the third angle in terms of the tangents of the other two angles, and would use the relation between geometric and arithmetic means.

ehild

 

[NeroKid]

tan(a)*tan(b)*tan(c) = -tan(b+c) *tan(b)*tan(c) = (tanb+tanc)/(1-tanb*tanc) *tanb*tanc
call S = tanb + tanc and P = tanb * tanc S^2>= 4P this should be easy from here on

 

[Michael_Light]

Still cannot do... how bout question (2)? Any hints?

 

[NeroKid]

you can use the langrange function to do it, it is quite easy if u use it

 

[ehild]

We can not help if you do not show any attempt.

ehild

 

[ehild]

you can use the langrange function to do it, it is quite easy if u use it

Note that it is Precalculus Math.

ehild

 

[NeroKid]

then just have to expand them to the sum and the product which is pretty much easier to solve

 

[rock.freak667]

For the second one I would consider a right angled triangle and see if that can help.

 

[tiny-tim]

1. Given a,b,c are acute angles and a + b + c =180. find the minimum of tan(a).tan(b).tan(c)

Following rock.freak's suggestion, I would write the tangent of the third angle in terms of the tangents of the other two angles, and would use the relation between geometric and arithmetic means.

ehild

tan(a)*tan(b)*tan(c) = -tan(b+c) *tan(b)*tan(c) = (tanb+tanc)/(1-tanb*tanc) *tanb*tanc
call S = tanb + tanc and P = tanb * tanc S^2>= 4P this should be easy from here on

Still cannot do...

Michael, are you trying?

You need to minimise (tanb+tanc)/(1-tanb*tanc) *tanb*tanc …

surely you have some idea how to do that?​