Memorization.
Were you expecting something a little more magic?
Just need a quick recap of certain things. Explanation has to be without the use of a calculator.
Using the unit circle, how do I find the sin and cos of pi/3.
Using the unit circle, how do I find the sin and cos of pi/6.
(I know the two have opposite cos and sin values, I just literally need to know how to get them without the use of a calculator but instead with the unit circle and the triangles.)
draw x and y coordinate axis
make the origin, the centre of circle (let it be o)
now draw a unit circle. let it x intercepts be a and -a
now raise the line segment oa at an angle of pi/3. drop a perpendiculat at x axis.
let point of intersection be c. since cosx=length of base/length of hypotenuse
but here hypotenuse =1 unit due to unit circle therefor
cosx=length of base therefor
cos(pi/3)=length of line segment oc=1/2
but this method is not actually useful as its not possible to measure the length of line every time for example value of cos(pi/6)=sqrt[3]/2 here sqrt[3] is an irrational number which is non terminating and non recurring so one will never be able to measure this length accurately.
for using them you may use other method for calculating values of sin,tan etc (not all values only for pi/2,pi/3,pi/4,pi/6, value for other angles can also be obtained but these values are elementary so can be obtained very easily using geometry)
or you may just memorize it