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# Tag Archives: General

## 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 .

## Equality on a triangle

Let be a triangle. Show that

Solution

Let denote the semiperimeter of the triangle. On account of the well known relations,

(1)

(2)

we have:

## Coincidences from … hell

Let denote the golden ratio. Prove that:

Solution

First of all we note that .

## On the geometrical view of an integral

Evaluate the integral

using geometric methods.

Solution

We are working on the following figure

Thus,

since the red angle is due to the triangle since ( ). Therefore , the green angle is . Finally, the area of the circular sector is equal to

where and .

## Trigonometric sum

Prove that

Solution

Consider the tridiagonal matrix . Its eigenvalues are . Hence,

and the result follows.

### Who is Tolaso?

Find out more at his Encyclopedia Page.