Question

Arrange these in order from least to greatest:

arctan(-sqrt3), arctan 0, arctan(1/2)

So far I got the first two values, arctan(-sqrt3), and that's 150 degrees. Arctan 0 would be zero degrees. I'll use just one answer for now, but I know there's more than one.

The last one I find a bit tricky, though, because it's not on the unit circle. I tried to use Pythagorean Theorem to get the third side (if triangles were involved) and got the square root of five. I can't use a calculator in this problem, but I checked anyway and found that arctan(1/2) is about 26.5 degrees.

Thus, I believe the order is arctan 0, arctan(1/2), and arctan(-sqrt3).

But is there any way I can figure out arctan(1/2) without the use of a calculator?

Thank you!

Answer 1

oops, i meant 120 degrees, not 150. my bad.

Answer 2

Your values are correct for a range of arctan from 0 to 180 degrees. Arctangents can take on values of arctan(x)±kπ due to its periodic nature. The period is 180 degrees.

If the range of the function is defined as from -90 to 90 degrees, the answer will be different.