How do you calculate a sine function in relation to the phase angle?

Answer 1

If A is your angle in radians, then
sin(A)=A-A^3/6+A^5/120-A^7/5040+..
In the exponents you have odd numbers. In the denominators you have factorials, so 5040=7!=7*6*5*4*3*2*1.
You can use this series to compute the sine of any angle to as much accuracy as you want (if you take enough terms in the sum).

Answer 2

You shift it left or right. For example, to find sin(x + pi/4), you know that sin(0) = 0. So set x + pi/4 = 0 and solve for x. If x = -pi/4, you get sin(0).

So sin(x + pi/4) = 0 for x = -pi/4.

You just shifted the sine curve to the left by pi/4.