Analysis

Approximating the Sinus function by polynomials (Lehman - Pesonen 2004-2016)

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Real functions: Sinus f(x) = sin x
Approximating the Sinus function by polynomial P(x) = a + b x + c x2 + d x3 around 0.
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Problem 1

Find the best values of a, b, c and d to approximate the sinus function around x = 0 by the polynomial P
P(x) = a + b x + c x2 + d x3.
The coefficient a =
The coefficient b =
The coefficient c =
The coefficient d =

Real functions: Sinus f(x) = sin x
Approximating the Sinus function by polynomial P(x) = a + b x + c x2 + d x3 around π/2.
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Problem 2

Find the best values of a, b, c and d to approximate the sine function around x = π/2 by the polynomial P
P(x) = a + b x + c x2 + d x3.
The coefficient a =
The coefficient b =
The coefficient c =
The coefficient d =

We saw from the Problem 2, that it is not at all an easy task with a polynomial of type P(x) = a + b x + c x2 + d x3, when the center of development is not 0.

The Taylor polynomial of degree n with center x0 of a one variable - enoughly many times differentiable - function f is

\sum_{k=0}^n \frac{f^{(k)}(x_0)}{k!} (x - x_0)^k
When x0 = 0, it is directly a polynomial of x and called Maclaurin polynomial. In the following 3 problems, find the best coefficients of the third degree Taylor polynomial approximation for Sinus, either by just trying to adjust the coefficient values in the dynamic figure, and also using calculations.
Real functions: Sinus f(x) = sin x
Approximating the Sinus function by polynomial P(x) = a + b(x - π/2) + c(x - π/2)2 + d(x - π/2)3 around π/2.
Note that:
π/2 = 1.571
π2/8 = 1.234
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Problem 3

Find the best values of a, b, c and d to approximate the sine function around x = π/2 by polynomial P
P(x) = a + b(x - π/2) + c(x - π/2)2 + d(x - π/2)3.

Try first this polynomial by choosing a = 0, b = 0, c = 1 and d = 0.
The coefficient a =
The coefficient b =
The coefficient c =
The coefficient d =

Real functions: Sinus f(x) = sin x
Approximating the Sinus function by polynomial P(x) = a + b x + c x2 + d x3 around π/2.
Note that:
π/2 = 1.571
π2/8 = 1.234
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Problem 2b

Find again (see Problem 2, but don't change the results you gave there!) the best values of a, b, c and d to approximate the sine function around x = π/2 by polynomial P
P(x) = a + b x + c x2 + d x3.
The coefficient a =
The coefficient b =
The coefficient c =
The coefficient d =

DO NEXT THE Problem 6! TRY THE TWO PROBLEMS 4 and 5 only if you have much time left!


Real functions: Sinus f(x) = sin x
Approximating the Sinus function by polynomial P(x) = a + b(x - 1) + c(x - 1)2 + d(x - 1)3 around x = 1.
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Problem 4

Find the best values of a, b, c and d to approximate the sine function around x = 1 by the polynomial P
P(x) = a + b(x - 1) + c(x - 1)2 + d(x - 1)3.
The coefficient a =
The coefficient b =
The coefficient c =
The coefficient d =

The Problem is even harder to do with polynomial centered at 0. Try the following!
Real functions: Sinus f(x) = sin x
Approximating the Sinus function by polynomial P(x) = a + b x + c x2 + d x3 around 1.
This page requires a Java capable browser. We also give you the following approximate numbers, can you imagine why?
1
6
 cos 1
≈
0.09005
sin 1 + 1
2
 cos 1
≈
1.112
1
2
 sin 1 − 5
6
 cos 1
≈
−0.02952
− 1
2
 sin 1 + 1
2
 cos 1
≈
−0.1506

Problem 5

Find the best values of a, b, c and d to approximate the sine function around x = 1 by the polynomial P
P(x) = a + b x + c x2 + d x3.
The coefficient a =
The coefficient b =
The coefficient c =
The coefficient d =

Real functions: Cosinus f(x) = cos x
Approximating the Cosinus function by polynomials at x = 0.
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Problem 6

Find the best values of a, b, c and d to approximate the cosine function around x = 0 by the polynomial P
P(x) = a + b x + c x2 + d x3.
The coefficient a =
The coefficient b =
The coefficient c =
The coefficient d =

6e. Can you see the connection between Problems 2, 3 and 6 ?




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Updated 17.3.2016