Graphing arccosine: What if we use different domains?

[srfriggen]

TL;DR Summary

What if, when restricting the domain for the cosine function to make it bijective, we use [180, 360 degrees].

Hello,

I'm a teacher and will be doing a lesson on "Graphing the inverse cosine function." In the lesson, I show the students a cosine function graphed from 0 to 360 degrees ( I use degrees to really drive home the point that this is a mapping between two different sets, namely angles and reals). I ask students to, "Look at the cosine graph (from 0 to 360 degrees) and find an interval that is 1-1 and onto." After that, we swap inputs and outputs to graph the arccos function.

There are obviously two correct answers: [0, 180] and [180, 360] (And infinitely many if you extend the original domain). But these two do have a difference... a specific example is the latter interval has 270 as an output which cannot be obtained from the first by an integer multiple of 360.

Can anyone help me find an intuitive and satisfying explanation to a high school student that would answer, "Mr. Fox, why can't we use [180, 360] when construction arccosine?

The way I understand it (and maybe I'm wrong) is that arccosine inputs a real number and outputs an angle, which is a measure of rotation. If we used [0, 180] to construct arccos then you can input 0 and output a rotation of 90 degrees from your starting point, 0 degrees. Now, say you constructed arccos from [180, 360] and input 0. Then your output would be 270, which is also a rotation of -90 degrees as long as your starting point is 360 degrees. So, for example, 30 degrees in the typical arccos would be equivalent to 330 degrees in the [180, 360] version, since 330 degrees is -30 degrees past 360. In either case, your inputs will be numbers from -1 to 1 and your outputs will be rotations past a certain starting point, but in the [180, 360] version you use negative angles.

I could be wrong and I want to be right so I'm asking for advice. Thank you.

 

[FactChecker]

When there are multiple choices like that, one is usually identified as the "primary principal branch". That is then the default standard. But the other branches must always be considered as valid alternatives if the problem allows those values.

 

[srfriggen]

When there are multiple choices like that, one is usually identified as the "primary branch". That is then the default standard. But the other branches must always be considered as valid alternatives if the problem allows those values.

And if a student asks, "Why can't we use the other branch?" How would you address that?

 

[fresh_42]

"You can, but if such a problem is related to a real world task or experiment, then reality will determine the range of possible angles."

 

[FactChecker]

And if a student asks, "Why can't we use the other branch?" How would you address that?

Those are certainly valid answers if one does not specify that the principal branch must be used and if they work in the given problem. Unless they are ruled out some way, they should be considered.

PS. Sorry, I should have said "principal branch", not "primary branch" in my prior response. I will correct it.

 

[srfriggen]

"You can, but if such a problem is related to a real world task or experiment, then reality will determine the range of possible angles."

Can you give a specific example?

 

[srfriggen]

Those are certainly valid answers if one does not specify that the principal branch must be used and if they work in the given problem. Unless they are ruled out some way, they should be considered.

PS. Sorry, I should have said "principal branch", not "primary branch" in my prior response. I will correct it.

I know what you mean by the principal branch. It's the same reason we restrict the domain for the square root function and call it the principal square root function. But for that function, we do it because we didn't know about imaginary numbers at the time. There are angles in the other branch that are completely valid, such as 270 degrees, but are being left out. It seems arbitrary. Why are the angles between 0 and 180 more important than the ones between 180 and 360?

 

[fresh_42]

Can you give a specific example?

Navigation comes to mind. Navigators calculate with full angles. I don't know enough about their specific work, but I could imagine that the inverse of a cosine could play a role for west bound vessels.

Another example could be calculations of the trace diagrams a particle accelerator produces.