# How do you evaluate # arc cos (1)#?

##### 2 Answers

arcos(1) = 0

#### Explanation:

If you take the cosine of an angle you have a numeric value. This number represents the ratio of length of particular sides of a right triangle.

When you evaluate "arccos" you are converting that ratio back to the original angle concerned. By the way, another way of writing "arcos" is

cosine is the ratio

Suppose you had

This is then 'converted' so that

However, people do not write the 1 so instead of

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

For this right triangle to exist the length of AB must be more than the length AC.

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

By writing arcos(1) in the question they are declaring that the length of DE is the same as the length of DF. For a right triangle this can only happen if

So arcos(1) = 0

#### Explanation:

The best way to define trigonometric functions and their inverse is by using the unit circle. Using this approach, we can define not only angles of right triangle, that are only measured in the interval

For detail description of this we can refer to UNIZOR.COM *Trigonometry* chapter of the course.

In short, the definitions are as follows.

Imagine a circle of a radius

An angle is formed by rotating a radius with an endpoint at *counterclockwise*. Thus, when the radius is directed vertically up (along the Y-axis towars its positive end), it's an angle of

When the radius is directed horizontally to the left towards negative X-coordinates, it's an angle of

For angles measured in negative numbers the rotation of the radius is *clockwise*.

By definition, for any angle

Inverse trigonometric function

Since **angle**