How to calculate cos, sine, etc. Without a calculator? I am currently studying computer science at university, and one of this semesters courses I am taking is linear algebra and calculus II. Every week there is a graded quiz, and a mid and end of semester exam. We are not allowed to use calculators in any of these. 
Today in the quiz, I failed to get an answer to one of the questions regarding the angle between two vectors, because I had to work out $\cos x = \frac2{2\sqrt\frac43}$. But I had no idea what angle that resulted in. 
I was just wondering if anyone had some tips as to how best to solve these problems in the future?
 A: Well, surely the first step is to write that as $\cos(x) = \frac{\sqrt{3}}{2}$. Could you have done it then?
At that point, it's about remembering a common triangle.
A: You can draw a unit circle and then calculate the trig values provided you can use a protractor+compass. Also, it is generally recommended one learns sin cos tan values for all  multiples of $15^\circ$.
Also, in your case, it can be greatly simplified as well... yielding :
$$\cos(x) = \frac{\sqrt{3}}{2} \Rightarrow x = 30^\circ$$
$30$ degrees is one solution.
Also... learning values of degrees 0,30,45,60,90 and 15 can make it easy to compute many problems you come across by using identities like sin(a+b) expansion, etc.
A: If you simplify what you got, it is $\frac{\sqrt{3}}{2}$.  There are several special angles related to a few triangles.  The triangles (in degrees) are the $[30-60-90][1]$ triangle and the $[45-45-90][2]$ (isoceles) triangle.  You can use these to calculate a number of cosines and sines using the unit circle.  For example, if you want $\sin\left(\frac{3\pi}{4}\right)$, you notice that this is $\frac{\pi}{4}$ radians beyond the $y$-axis.  Thus you can embed a $45-45-90$ triangle against the $y$-axis and calculate values.  Here is a video.
