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

• Michael_Light
In summary, I don't know how to solve the first homework problem and I'm not sure how to solve the second.
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) >= 31/2

## The Attempt at a Solution

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

Last edited:
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°)

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... how bout question (2)? Any hints?

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

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

ehild

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

Note that it is Precalculus Math.

ehild

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

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

Michael_Light said:
1. Given a,b,c are acute angles and a + b + c =180. find the minimum of tan(a).tan(b).tan(c)
ehild said:
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 said:
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 said:
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?​

## 1. What is Trigonometry?

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles.

## 2. What is the minimum value of tan(a).tan(b).tan(c)?

The minimum value of tan(a).tan(b).tan(c) is 0. This occurs when one of the angles (a, b, or c) is equal to 0 degrees.

## 3. How can I find the minimum value of tan(a).tan(b).tan(c)?

To find the minimum value of tan(a).tan(b).tan(c), you can use the trigonometric identities and properties such as the fact that tan(0) = 0 and the product-to-sum identity for tangent.

## 4. Why is finding the minimum value of tan(a).tan(b).tan(c) useful?

Finding the minimum value of tan(a).tan(b).tan(c) can be useful in solving optimization problems, where the goal is to minimize a certain quantity. It can also be used in trigonometric equations and applications involving triangles.

## 5. Can the minimum value of tan(a).tan(b).tan(c) be negative?

No, the minimum value of tan(a).tan(b).tan(c) cannot be negative. This is because the tangent function is always positive in the first and third quadrants, and when one of the angles is equal to 0 degrees, the product of the tangents will be 0.

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