Instead of memorizing the values you can learn the shape of the graphs and/or learn the meaning of sin/cos on a right angle triangle.
I think I got ok through a math/comp.sci. degree without memorizing values but I will say the "mechanical" aspects of math including trig functions and things like integrals and derivatives involve drilling those enough to make these things automatic (which is sort of memorizing but not exactly). I was always lazy so I never got really good at that (and my calculus related grades are evidence of that ;) ).
I think I got ok through a math/comp.sci. degree without memorizing values but I will say the "mechanical" aspects of math including trig functions and things like integrals and derivatives involve drilling those enough to make these things automatic (which is sort of memorizing but not exactly). I was always lazy so I never got really good at that (and my calculus related grades are evidence of that ;) ).