Chapter 4.5, Problem 95E

a.

To determine

Is the error in the approximation the same in each case? Explain.

Answer to Problem 95E

  $\boxed{\text{Approximation Error}=0}$

Explanation of Solution

Given information:

Use the polynomial approximations of the sine and cosine functions in Exercise  94 to approximate each function value. Compare the results with those given by a calculator. Is the error in the approximation the same in each case? Explain.

  $\sin \frac{1}{2}$

Calculation:

Approximation of sine function by polynomial is,

  $\sin x\approx x-\frac{x^3}{3!}+\frac{x^5}{5!}$

Using approximation above, for $x=\frac{1}{2}$ , you get

  $\sin \frac{1}{2}\approx \left(\frac{1}{2}\right)-\frac{{\left(\frac{1}{2}\right)}^3}{3!}+\frac{{\left(\frac{1}{2}\right)}^5}{5!}$

  $=\frac{1}{2}-\frac{1}{8\left(6\right)}+\frac{1}{32\left(120\right)}$

  $=0.479427083$

  $\approx \boxed{0.4794}$

Using graphing calculator, you get

  $\sin \frac{1}{2}=0.4794255386$

  $\approx \boxed{0.4794}$

Hence, the Approximation error is $0$

b.

To determine

Is the error in the approximation the same in each case? Explain.

Answer to Problem 95E

  $\boxed{\text{Approximation error=0}\text{.0002}}$

Explanation of Solution

Given information:

Use the polynomial approximations of the sine and cosine functions in Exercise  94 to approximate each function value. 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,

  $\sin x\approx x-\frac{x^3}{3!}+\frac{x^5}{5!}$

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

  $\sin 1\approx \left(1\right)-\frac{{\left(1\right)}^3}{3!}+\frac{{\left(1\right)}^5}{5!}$

  $=1-\frac{1}{6}+\frac{1}{120}$

  $=0.8416666667$

  $\approx \boxed{0.8415}$

Using graphing calculator , you get

  $\sin 1=0.8414709848$

  $\approx \boxed{0.8415}$

Hence, the Approximation error is $0.0002$ .

c.

To determine

Is the error in the approximation the same in each case? Explain.

Answer to Problem 95E

  $\boxed{\text{Approximation error=0}}$

Explanation of Solution

Given information:

Use the polynomial approximations of the sine and cosine functions in Exercise  94 to approximate each function value. Compare the results with those given by a calculator. Is the error in the approximation the same in each case? Explain.

  $\sin \frac{\pi }{6}$

Calculation:

Approximation of sine function by polynomials is,

  $\sin x\approx x-\frac{x^3}{3!}+\frac{x^5}{5!}$

Using approximation above, for $x=\frac{\pi }{6}$ , you get

  $\sin \frac{\pi }{6}\approx \left(\frac{\pi }{6}\right)-\frac{{\left(\frac{\pi }{6}\right)}^3}{3!}+\frac{{\left(\frac{\pi }{6}\right)}^5}{5!}$

  $=\frac{\pi }{6}-\frac{{\pi }^3}{216\left(6\right)}+\frac{{\pi }^5}{7776\left(120\right)}$

  $=0.5000021326$

  $\approx 0.5000$

Using graphing calculator, you get

  $\sin \frac{\pi }{6}=0.5$

Hence, the Approximation error is $0$ .

d.

To determine

Is the error in the approximation the same in each case? Explain.

Answer to Problem 95E

  $\boxed{\text{Approximation error= }0}$

Explanation of Solution

Given information:

Use the polynomial approximations of the sine and cosine functions in Exercise  94 to approximate each function value. 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,

  $\cos x\approx 1-\frac{x^2}{2!}+\frac{x^4}{4!}$

Using approximation above, for $x=-0.5$ , you get

  $\cos \left(-0.5\right)\approx 1-\frac{{\left(-0.5\right)}^2}{2!}+\frac{{\left(-0.5\right)}^4}{4!}$

  $=1-\frac{0.25}{2}+\frac{0.0625}{24}$

  $=0.8776041667$

  $\approx \boxed{0.8776}$

Using graphing calculator, you get

  $cos\left(-0.5\right)=0.8775825619$

  $\approx \boxed{0.8776}$

Hence, the $\text{Approximation error is }0$

e.

To determine

Is the error in the approximation the same in each case? Explain.

Answer to Problem 95E

  $\boxed{\text{Approximation error=0}\text{.0014}}$

Explanation of Solution

Given information:

Use the polynomial approximations of the sine and cosine functions in Exercise  94 to approximate each function value. 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,

  $\cos x\approx 1-\frac{x^2}{2!}+\frac{x^4}{4!}$

Using approximation above, for $x=-0.5$ , you get

  $\cos 1\approx 1-\frac{{\left(1\right)}^2}{2!}+\frac{{\left(1\right)}^4}{4!}$

  $=1-\frac{1}{2}+\frac{1}{24}$

  $=0.5416666667$

  $\begin{array}{l}\approx \boxed{0.5417}\\ \end{array}$

Using graphing calculator, you get

  $\cos 1=0.5403023059$

  $\approx \boxed{0.5403}$

Approximation error is $0.0014$