First, as integral told you, and you apparently did not understand because you immediately asked the same question again, is that there is NO way to "know that 0.866 is actually square root 3/2" because it is not true! .866 is approximately square root of 3, divided by 2. And the only way to know that is to actually take the square root of 3 to four decimal places, divide by 2, and round to three decimal places.
As for the angles Mark44 mentions- If one angle of a right triangle is 45 degrees (\pi/4 radians), then the other angle must be 90- 45= 45 degrees also. That means that the right triangle is "isosceles"- if two angles are the same, then the two legs are the same length. Taking that length to be 1, by the Pythagorean theorem, the hypotenuse has length \sqrt{2} and it is easy to see that the sin(45)= 1/\sqrt{2}= \sqrt{2}/2.
An equilateral triangle, with all sides the same length, say, 1, must have all angles the same length: 180/2= 60 degrees or \pi/3 radians. If you drop a perpendicular from one vertex to the opposite side, it is easy to show that both the opposite side and the angle are bisecte so you have two right triangles with angles 60 degrees and 30 degrees (\pi/6 radians). The hypotenuse has length 1 and the side opposite the 30 degree angle has length 1/2. You can then use the Pythagorean theorem to show that the other leg, opposite the 60 degree angle, has length \sqrt{3}/2.
That is enough to tell you that sin(30)= 1/2, cos(30)= \sqrt{3}/2, sin(60)= \sqrt{3}/2, and cos(60)= 1/2.
