sine , cosine , tangent

The answers below assumes that the angle "X" is measured in RADIANS, which is a real number, typically between 0 and 2*pi, with 2*Pi = 360 degree.

You need the transformation from degrees to radian, using the following:
X is degree, Y is radians
X/360 = Y/2*pi
So Y = pi*X/180

Then we can apply the Taylor series:
sin(Y) = Y-Y^3/3! + Y^5/5! - Y^7/7! + ...
cos(Y)= 1-Y^2/2! + Y^4/4! - Y^6/6! + ...
tan(Y) = sin(Y)/cos(Y)

This is how your calculator actually calculate the numerical value of sin/cos/tan, where the sum is truncated when the accuracy is high enough (the additional term does not change the answer significantly)

I understand that the calculator would not calculate the value of sin, cos or tan directly by Taylor series but use some other quicker formula, typically a polynomial of order n, to approximate the final answer.

Taylor's Series:

sinx = x - (x^3)/3! + (x^5)/5! - (x^7)/7! + (x^9)/9! - (x^11)/11! +..... where x in radian

cosx = 1 - (x^2)/2! + (x^4)/4! - (x^6)/6! +......

See the site for more details.... Thanks

sin(x) = x - (x^3)/3! + (x^5)/5! - (x^7)/7! + ....... for all x
cos(x) = 1 - (x^2)/2! + (X^4)/4! - (x^6)/6! + ...... for all x
tan(x) = sin(x)/cos(x)