Is there a more precise way to determine the percentage of actual length of a projected segment then by measuring? I was hoping for a formula to find out what percentage of a length projects onto a line. The projection lines are perpendicular, like in orthographic projections. 

I’m not good at mathematics, so I measured some angles, but even so, it’s not precise enough. Here are my measuring results: 
10° - 98.9%; 20° - 93.8%; 30° - 87.3%; 40° - 76.7%; 50° - 64.7%; 60° - 50%; 70° - 34.9%; 80° - 18.1%. 
Aside that, I can’t measure every angle and it would be hard to estimate something like 38°.
Also, is there a way to determine at what rate a projection grows or shrinks at an equal angle of rotation? 
 A: You need the cosine function.
$$\cos10°=0.98480775301220805936674302458952\cdots,\\
\cos20°=0.93969262078590838405410927732473\cdots,\\
\cos30°=0.86602540378443864676372317075294\cdots,\\\cdots
$$
You find that on any scientific calculator. Make sure to set the angle mode to degrees.
Note that $\cos30°$ is exactly $\dfrac{\sqrt3}2$.
A: The projected length is the actual length times the cosine of the angle between that direction and the line of projection. So you'd have $\cos(10°)\approx0.9848$, $\cos(20°)\approx0.9397$, $\cos(30°)\approx0.8660$ and so on. Notably $\cos(60°)=0.5$ is exact.
This use of the cosine follows from the geometric definition of that function: the cosine of an angle in a right-angled triangle is the length of the edge between that angle and the right angle divided by the length of the edge opposite the right angle. In your case, the edge opposite the right angle is the real length, while the edge between input angle and right angle is the projected length.
When you ask about growth or shrinkage of that function as the angle changes, you are essentially asking for the derivative: given an inifinitesimal change in input, how does the output change in response? What's the ratio of output change divided by input change? The derivative of $\cos(t)$ is
$$\frac{\mathrm d}{\mathrm dt}\cos(t)=-\sin(t)$$
i.e. the change is proportional to the sine of the angle, and has opposite sign. I say proportional not equal because you might want to take the speed of rotation into account. The change would be exactly to $-\sin$ of the angle if you were rotating at a speed of one rotation every $2\pi\approx6.2832$ units of time.
The number $\pi\approx3.1416$ appears in the above formula because for many applications, it is easier to measure angles not in degrees but in radians. This is expressing angles as the ratio of the length of a circular arc divided by the radius, and for a full circle that ratio would be $2\pi$. For the derivative rule I stated, having angles measured in radians means both the input and the output of the function is a number (and actually a ratio), which is necessary to to compare them. Otherwise you'd end up dividing change in length ratio (which is a number) by change in angle (which is an angle) and without a means of expressing angles as numbers, that computation wouldn't make sense.
