Quick guess on the cosine estimate is that the student is reversing sine and cosine since sin 270 = -1. I might be generous here…

The distance question? Oh man, my life would be much better if I understood the multitude of ways that students misunderstand distance. I think the intent of the question is to get students to make the connection to the unit circle values that they (kind of) know. If anything, I’d superficially guess that students think that (1, 1) is 2 units from the origin rather than 1 unit (because, you know, it takes two steps) so, sadly, I don’t have any insights into this part of the question.

I agree with mrdardy’s suggestion about the student potentially reversing sine and cosine in their answer to the question. If it was sin(260), -5/6 would be a reasonable estimation.

When it comes to the distance question, I’m not sure exactly what the student’s thought was. Some students might say when pressed that the point (1,1) is 1 unit from the origin if they didn’t really take time to consider.

On the distance question, I’d say that this is an example where a picture is worth a thousand words. With the picture, they could easily see that the distances from the points they provide are greater than those mentioned. One problem with the picture is that it might lead students to “connect the dots” drawing a square instead of a circle and still getting the wrong answer.

Tom Chandler

I have a hunch about the distance question. Students are used to looking for patterns. A student could have looked at the examples given — (0, 1), (1, 0), (-1, 0), (0, -1) — and reason that similar combinations should also work: (1, -1), (1, 1). The student probably never paused to consider the reason why the four points given are 1 unit from the origin.