sin(x) = a_{0} + a_{1}x + a_{2}x^{2} + a_{3}x^{3} + …  (1)

cos(x) = a_{1} + 2a_{2}x + 3a_{3}x^{2} + …  (2)

cos(0) = a_{1}  (3)

−sin(x) = 2a_{2} + 6a_{3}x + …  (4)

−cos(x) = 6a_{3} + …  (5)

sin(x) = x − \frac{1}{3!}x^{3} + …  (6)

a_{k} = \frac{1}{k!} \frac{d^{k}f(x)}{dx^{k}}|_{x=0}  (7)

f (x) = f (x_{0}) + (x − x_{0}) f′|_{x_{0}} + \frac{1}{2} (x − x_{0})^{2} f′′|_{x_{0}} + \frac{1}{3!} (x − x_{0})^{3} f′′′|_{x_{0}} + …

(8)

f_{linear}(x) = a + b(x − x_{0})  (9)

a_{0} = a
a_{1} = b  (10)

f_{quadrtic}(x) = a + b(x − x_{0}) + c(x − x_{0})^{2}  (11)

a_{0} = a
a_{1} = b
a_{2} = c  (12)

P_{n}(x) = ∑_{k=0}^{n} \frac{f^{(k)}(x_{0})}{k!}(x − x_{0})^{k}  (13)

R_{n}(x) = \frac{f^{(n+1)}(c)}{(n + 1)!}(x − x_{0})^{n+1}  (14)

f(x) = P_{n}(x) + R_{n}(x)  (15)