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# Tag Archives: Inequality

## Trigonometric inequality on a triangle

Let be a triangle. Prove that

Solution

The function is convex, thus:

The result follows. In fact, something a bit stronger holds. Let denote the inradius , the circumradius and the semiperimeter. Then,

Indeed,

in view of the known identities

(1)

(2)

(3)

## Inequality involving area of a triangle

Show that in any triangle with area the following holds:

Solution

Let is the semiperimeter , the circumradius and the inradius. From the law of sines we find

(1)

as well as

(2)

Now,

Substitute the preceding equalities into the last inequality  and simplifying we obtain

From

the last inequality becomes

which is true by the rearrangement inequality.

## Trigonometric inequality on an acute triangle

Prove that in any acute triangle the following inequality holds:

Solution

Since it holds that

(1)

and thus

(2)

Using Nesbitt’s inequality we see that

Equality holds if-f .

## Trigonometric inequality

Prove that the following inequality holds in any triangle:

Solution

Let denote the semiperimeter of the triangle. Using the cosine theorem we have that

from which it follows that

(1)

(2)

(3)

Thus, by Cauchy’s inequality we have:

Hence,

## Inequality of a function

Let be a differentiable function with continuous derivative. Prove that:

Solution

For it holds that

Taking absolute values and using basic properties of the integral we get

Integrating we have:

(1)

Working similarly on we get

(2)

Adding equations we get the result.

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