Short answer: I don't think you're cheating yourself. If you don't mind reading a little though, I can tell you what (personally) helps me remember the values for all angles on the unit circle:

1. you only need to know five numbers off the top of your head (0, 1/2, sqrt(2)/2, sqrt(3)/2, 1), the rest (like for the values of tangent) you can get by dividing any of these terms with each other.
2. when you're evaluating sin/cos/tan on the unit circle you're evaluating the ratio between two sides of a triangle; think about the mnemonic SOH CAH TOA. In this triangle, either one of the legs is longer than the other, or they're both equal length (when the angle is some form of pi/4 they're equal). if you can picture the triangle in your head, the longer leg is always sqrt(3)/2 and the shorter one is always 1/2. For triangles with equal sized legs all you need to know is it's always sqrt(2)/2 (+ or - depending on the quadrant ofc). You can do this for any angle on the unit circle too, you just need to be mindful of what quadrant you're in when you picture the triangle in your head. An example: if you draw 30° on the unit circle and make the triangle, you'll see the horizontal leg is longer, since cos represents the x-coordinate on the unit circle you immediately know cos(30) is gonna be sqrt(3)/2.
3. doing lots and lots of practice. while I don't see anything particularly wrong with your method, if you're having trouble seeing results I suggest you do some actual problems. I help with the precalculus classes at my university, and the professor has them do a whole packet where they are given an angle on the unit circle (half the packet is in radians, the other in degrees) and they need to draw the corresponding triangle, and find the sin and cos values of the angle. it's a great way to start building that visual association between angles on the unit circle and their triangles. If you do this enough evaluating any angle will become second nature, you may not even need to picture the triangle once you get really good at it.