## Arcosine function

Given a function such that

(1)

1. Evaluate .
2. Prove that is one to one.
3. Prove that for all .
4. Find the range of .
5. Sketch the graph of .

Solution

1. Setting at we have that

However,

Thus .

2. Let such that . Thus,

Hence is .

3. Setting we have

since is strictly decreasing in .

4. because .
5. The graph is seen at the following figure:

Let . Prove that

## Hyperbolic Triangle

The vertices of a triangle lie on the hyperbola . Prove that its orthocentre also lies on the hyperbola.

Solution

We are working on the following figure

Let , and . Let us denote as its orthocentre. We have that:

Hence, the slope of the altitude is . Similarly,

Hence, the slope of the altitude is . Hence,

and

Solving this linear system we have

and finally . So, . This proves the claim.

## Limit of radius of inscribed circle

Consider the points ,   with . Let be the radius of the inscribed circle of the triangle .

Prove that

Solution

Since we deduce that as . Since the incenter lies on the bisector of , it follows that if is the projection of on the axis

\rho = AD \tan \frac {A}{2}\leq 1\tan \frac {A}{2} \rightarrow 0

## Limit

Evaluate the limit

Solution

Rewrite the limit as

Using the definition of the derivative we get that limit equals to