# How do you tell whether the value of sin (-45 degrees) is positive, negative, zero or undefined?

Sep 25, 2015

First of all, the sine function is defined all over the real set, so there's no chanche that \sin(-45°) is undefined.
Now, you can solve the problem in a geometrical way: since the sine of an angle is the $y$-coordinate of the point of the unit circle related to that angle, you can clearly see that angles from 0° to 180° have a non-negative sine value, while angles from 180° to 360° have a non-positive value. Since an angle of -45° describes the same point as an angle of 360-45=315°, we are in the second case, and the sine is positive.
If you want to see it using the definition, the point in the unit circle identified by a -45° angle is $\left(\frac{\sqrt{2}}{2} , - \frac{\sqrt{2}}{2}\right)$, so its $y$-coordinate is negative.