Different ways (and the fastest) to compute sines (and cosines) in Arduino
The two basic methods are mathematical calculation (with polynomials) and lookup tables. 

The Arduino's math library (libm, part of avr-libc) uses the former. It is optimised for the AVR in that it is written with 100% assembly language, and as such is almost impossible to follow what it's doing (there's zero comments as well). Rest assured though it will be the most optimised pure-float implementation brains far superior to ours could come up with.

However the key there is float. Anything on the Arduino involving floating point is going to be heavyweight by comparison to pure integer, and since you are only requesting integers between 0 and 90 degrees a simple lookup table is by far the simplest and most efficient method.

A table of 91 values will give you everything from 0 to 90 inclusive. However if you make that a table of floating point values between 0.0 and 1.0 you still then have the inefficiency of working with floats (granted not as inefficient as calculating sin with floats), so storing a fixed point value instead would be far more efficient.

That may be as simple as storing the value multiplied by 1000, so you have between 0 and 1000 instead of between 0.0 and 1.0 (for instance sin(30) would be stored as 500 instead of 0.5). More efficient would be to store the values as, for instance, a Q16 value where each value (bit) represents 1/65536th of 1.0. These Q16 values (and the related Q15, Q1.15, etc) are more efficient to work with since you have powers-of-two which computers love to work with instead of powers-of-ten which they hate working with.

Don't forget as well that the sin() function expects radians, so you first have to convert your integer degrees into a floating point radians value, making the use of sin() even more inefficient compared to a lookup table that can work directly with the integer degrees value.

A combination of the two scenarios, though, is possible. Linear interpolation will allow you to get an approximation of a floating point angle between two integers. It's as simple as working out how far between two points in the lookup table you are and creating a weighted average based on that distance of the two values. For instance if you are at 23.6 degrees you take (sintable[23] * (1-0.6)) + (sintable[24] * 0.6). Basically your sine wave becomes a series of discrete points joined together by straight lines.  You trade accuracy for speed.