How do you evaluate # arc cos (1)#?
arcos(1) = 0
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
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
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