Easy way to keeping angles between -179 and 180 degrees

// reduce the angle  
angle =  angle % 360; 

// force it to be the positive remainder, so that 0 <= angle < 360  
angle = (angle + 360) % 360;  

// force into the minimum absolute value residue class, so that -180 < angle <= 180  
if (angle > 180)  
    angle -= 360;  

Try this instead!

atan2(sin(angle), cos(angle))

atan2 has a range of [-π, π). This takes advantage of the fact that tan θ = sin θ / cos θ, and that atan2 is smart enough to know which quadrant θ is in.

Since you want degrees, you will want to convert your angle to and from radians:

atan2(sin(angle * PI/180.0), cos(angle * PI/180.0)) * 180.0/PI

Update My previous example was perfectly legitimate, but restricted the range to ±90°. atan2's range is the desired value of -179° to 180°. Preserved below.

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Try this:

asin(sin(angle)))

The domain of sin is the real line, the range is [-1, 1]. The domain of asin is [-1, 1], and the range is [-PI/2, PI/2]. Since asin is the inverse of sin, your input isn't changed (much, there's some drift because you're using floating point numbers). So you get your input value back, and you get the desired range as a side effect of the restricted range of the arcsine.

Since you want degrees, you will want to convert your angle to and from radians:

asin(sin(angle * PI/180.0)) * 180.0/PI

(Caveat: Trig functions are bazillions of times slower than simple divide and subtract operations, even if they are done in an FPU!)