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srfriggen

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- 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

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.

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.

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