Chapter 4.5, Problem 101E

a.

To determine

Find approximation error of the function.

Answer to Problem 101E

  $\boxed{0}$

Explanation of Solution

Given information:

Use the polynomial approximations of the sine and cosine functions in Exercise 100 to approximate the following function values. Compare the results with those given by a calculator. Is the error in the approximation the same in each case? Explain.

  $\sin \begin{array}{c}1\\ 2\end{array}$

Calculation:

Approximation of sine function by polynomials is,

  $\sin x\approx x-\begin{array}{c}{x}^3\\ 3!\end{array}+\begin{array}{c}{x}^5\\ 5!\end{array}$

Using approximation above, for $x=\begin{array}{c}1\\ 2\end{array}$ , we get,

  $\sin \begin{array}{c}1\\ 2\end{array}\approx \left(\begin{array}{c}1\\ 2\end{array}\right)-\begin{array}{c}{\left(\begin{array}{c}1\\ 2\end{array}\right)}^3\\ 3!\end{array}+\begin{array}{c}{\left(\begin{array}{c}1\\ 2\end{array}\right)}^5\\ 5!\end{array}$

  $\begin{array}{l}=\begin{array}{c}1\\ 2\end{array}-\begin{array}{c}1\\ 8\left(6\right)\end{array}+\begin{array}{c}1\\ 32\left(120\right)\end{array}\\ =0.4794255386\\ \approx 0.4794\end{array}$

Using graphic calculator, you get

  $\begin{array}{l}\sin \begin{array}{c}1\\ 2\end{array}=0.4794255386\\ \approx 0.4794\end{array}$

Hence, approximation error is $\boxed{0}$ .

b.

To determine

Find approximation error of the function.

Answer to Problem 101E

  $\boxed{0.0002}$

Explanation of Solution

Given information:

Use the polynomial approximations of the sine and cosine functions in Exercise 100 to approximate the following function values. Compare the results with those given by a calculator. Is the error in the approximation the same in each case? Explain.

  $\sin 1$

Calculation:

Approximation of sine function by polynomials is,

  $\sin x\approx x-\begin{array}{c}{x}^3\\ 3!\end{array}+\begin{array}{c}{x}^5\\ 5!\end{array}$

Using approximation above, for $x=1$ , we get,

  $\sin 1\approx \left(1\right)-\begin{array}{c}{\left(1\right)}^3\\ 3!\end{array}+\begin{array}{c}{\left(1\right)}^5\\ 5!\end{array}$

  $\begin{array}{l}=1-\begin{array}{c}1\\ 6\end{array}+\begin{array}{c}1\\ 120\end{array}\\ =0.8416666667\\ \approx 0.8417\end{array}$

Using graphic calculator, you get

  $\begin{array}{l}\sin 1=0.8414709848\\ \approx 0.8415\end{array}$

Hence, approximation error is $\boxed{0.0002}$ .

c.

To determine

Find approximation error of the function.

Answer to Problem 101E

  $\boxed{0}$

Explanation of Solution

Given information:

Use the polynomial approximations of the sine and cosine functions in Exercise 100 to approximate the following function values. Compare the results with those given by a calculator. Is the error in the approximation the same in each case? Explain.

  $\sin \begin{array}{c}\pi \\ 6\end{array}$

Calculation:

Approximation of sine function by polynomials is,

  $\sin x\approx x-\begin{array}{c}{x}^3\\ 3!\end{array}+\begin{array}{c}{x}^5\\ 5!\end{array}$

Using approximation above, for $x=\begin{array}{c}\pi \\ 6\end{array}$ , we get,

  $\sin \begin{array}{c}\pi \\ 6\end{array}\approx \left(\begin{array}{c}\pi \\ 6\end{array}\right)-\begin{array}{c}{\left(\begin{array}{c}\pi \\ 6\end{array}\right)}^3\\ 3!\end{array}+\begin{array}{c}{\left(\begin{array}{c}\pi \\ 6\end{array}\right)}^5\\ 5!\end{array}$

  $\begin{array}{l}=\left(\begin{array}{c}\pi \\ 6\end{array}\right)-\begin{array}{c}{\pi }^3\\ 216\left(6\right)\end{array}+\begin{array}{c}{\pi }^5\\ 7776(120)\end{array}\\ =0.5000021326\\ \approx 0.5000\end{array}$

Using graphic calculator, you get

  $\sin \frac{\pi }{6}=0.5$

Hence, approximation error is $\boxed{0}$ .

d.

To determine

Find approximation error of the function.

Answer to Problem 101E

  $\boxed{0}$

Explanation of Solution

Given information:

Use the polynomial approximations of the sine and cosine functions in Exercise 100 to approximate the following function values. Compare the results with those given by a calculator. Is the error in the approximation the same in each case? Explain.

  $\cos \left(-0.5\right)$

Calculation:

Approximation of cosine function by polynomials is,

  $\cos x\approx x-\begin{array}{c}{x}^2\\ 2!\end{array}+\begin{array}{c}{x}^4\\ 4!\end{array}$

Using approximation above, for $x=\left(-0.5\right)$ , we get,

  $\cos \left(-0.5\right)\approx \left(-0.5\right)-\begin{array}{c}{\left(-0.5\right)}^2\\ 2!\end{array}+\begin{array}{c}{\left(-0.5\right)}^4\\ 4!\end{array}$

  $\begin{array}{l}1-\begin{array}{c}0.25\\ 2\end{array}+\begin{array}{c}0.0625\\ 24\end{array}\\ =0.8776041667\\ \approx 0.8776\end{array}$

Using graphic calculator, you get

  $\begin{array}{l}\cos \left(-0.5\right)=0.8775825619\\ \approx 0.8776\end{array}$

Hence, approximation error is $\boxed{0}$ .

e.

To determine

Find approximation error of the function.

Answer to Problem 101E

  $\boxed{0.0014}$

Explanation of Solution

Given information:

Use the polynomial approximations of the sine and cosine functions in Exercise 100 to approximate the following function values. Compare the results with those given by a calculator. Is the error in the approximation the same in each case? Explain.

  $\cos 1$

Calculation:

Approximation of cosine function by polynomials is,

  $\cos x\approx x-\begin{array}{c}{x}^2\\ 2!\end{array}+\begin{array}{c}{x}^4\\ 4!\end{array}$

Using approximation above, for $x=1$ , we get,

  $\cos 1\approx 1-\begin{array}{c}{1}^2\\ 2!\end{array}+\begin{array}{c}{1}^4\\ 4!\end{array}$

  $\begin{array}{l}1-\begin{array}{c}1\\ 2\end{array}+\begin{array}{c}1\\ 24\end{array}\\ =0.541666667\\ \approx 0.5417\end{array}$

Using graphic calculator, you get

  $\begin{array}{l}\cos 1=0.5403023059\\ \approx 0.5403\end{array}$

Hence, approximation error is $\boxed{0.0014}$ .

f.

To determine

Find approximation error of the function.

Answer to Problem 101E

  $\boxed{0.0003}$ .

Explanation of Solution

Given information:

Use the polynomial approximations of the sine and cosine functions in Exercise 100 to approximate the following function values. Compare the results with those given by a calculator. Is the error in the approximation the same in each case? Explain.

  $\cos \frac{\pi }{4}$

Calculation:

Approximation of cosine function by polynomials is,

  $\cos x\approx x-\begin{array}{c}{x}^2\\ 2!\end{array}+\begin{array}{c}{x}^4\\ 4!\end{array}$

Using approximation above, for $x=1$ , we get,

  $\cos \frac{\pi }{4}\approx \frac{\pi }{4}-\begin{array}{c}{\left(\frac{\pi }{4}\right)}^2\\ 2!\end{array}+\begin{array}{c}{\left(\frac{\pi }{4}\right)}^4\\ 4!\end{array}$

  $\begin{array}{l}1-\begin{array}{c}{\pi }^2\\ \left(16\right)2\end{array}+\begin{array}{c}{\pi }^4\\ \left(256\right)24\end{array}\\ =0.7074292067\\ \approx 0.7074\end{array}$

Using graphic calculator, you get

  $\begin{array}{l}\cos \frac{\pi }{4}=0.7071067812\\ \approx 0.7074\end{array}$

Approximation error is $\boxed{0.0003}$ .

Hence , approximation errors are not same in each case.