I’d like to discuss a common mistake students make in trigonometry… as well as the one-liner that I use to (hopefully) help students not make this mistake in the future.

Question. Find all solutions (rounded to the nearest tenth of a degree) of .

Erroneous Solution. Plugging into a calculator, we find that .

I offer a thought bubble if you’d like to think about why this answer is wrong.

The student correctly found the unique angle between and so that . That’s the definition of the arcsine function. However, there are plenty of other angles whose sine is equal to . This can happen in two ways.

First, if $\sin x > 0$, then the angle could be in either the first quadrant or the second quadrant (thanks to the mnemonic All Scholars Take Calculus). So could be (accurate to one decimal place) equal to either or else . Students can visualize this by drawing a picture, talking through each step of its construction (first black, then red, then brown, then green, then blue).

However, most students really believe that there’s a second angle that works when they see the results of a calculator.

Second, any angle that’s coterminal with either of these two angles also works. This can be drawn into the above picture and, as before, confirmed with a calculator.

So the complete answer (again, approximate to one decimal place) should be and $126.9 + 360n^{\circ}$, where is an integer. Since integers can be negative, there’s no need to write in the solution.

Therefore, the student who simply answers has missed infinitely many solutions. The student has missed every nontrivial angle that’s coterminal with and also every angle in the second quadrant that also works.

Here’s my one-liner — which never fails to get an embarrassed laugh — that hopefully helps students remember that merely using the arcsine function is not enough for solving problems such as this one.

You’ve forgotten infinitely many solutions. So how many points should I take off?

I'm a Professor of Mathematics and a University Distinguished Teaching Professor at the University of North Texas. For eight years, I was co-director of Teach North Texas, UNT's program for preparing secondary teachers of mathematics and science.
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