Chapter 4.3, Problem 77E

Solution

To fill: The missing values in the table to show that $\sin \theta \approx \theta$ (in radians). It is given that Physicists often use the approximation $\sin \theta \approx \theta$ for small values of ?. For what values of ? is the magnitude of the error in approximating sin ? by ? less than $1\%$ of sin $\theta$ ?

  $\begin{array}{ccc}\theta & \sin \theta & \sin \theta -\theta \\ -0.03& & \\ -0.02& & \\ -0.01& & \\ 0& & \\ 0.01& & \\ 0.02& & \\ 0.03& & \end{array}$

For $\left|\theta \right|<0.2441$ error in approximating $\sin \theta \text{ by }\theta$ less than $1\%$ of $\sin \theta$ .

The complete table is shown below:

  $\begin{array}{cccc}\theta & \sin \theta & \sin \theta -\theta & \frac{|\sin \theta -\theta |}{\left|\sin \theta \right|}\\ -0.03& -0.029996& 4.4998\times {10}^{-6}& 0.00015\\ -0.02& -0.019999& 1.33331\times {10}^{-6}& 0.00006667(.006)\\ -0.01& -0.0099998& 1.66666\times {10}^{-7}& 0.000016667\\ 0& 0& 0& 0\\ 0.01& 0.0099998& -1.66666\times {10}^{-7}& 0.000016667\\ 0.02& 0.019999& -1.33331\times {10}^{-6}& 0.00006667\\ 0.03& 0.029996& -4.4998\times {10}^{-6}& 0.00015\end{array}$

Given information:

It is given that $\sin \theta \approx \theta$ when $\left|\theta \right|$ is small.

The equation is $\frac{\left|\sin \theta -\theta \right|}{\left|\sin \theta \right|}<0.01$ .

Explanation:

Consider the given information:

The complete table is:

  $\begin{array}{cccc}\theta & \sin \theta & \sin \theta -\theta & \frac{|\sin \theta -\theta |}{\left|\sin \theta \right|}\\ -0.03& -0.029996& 4.4998\times {10}^{-6}& 0.00015\\ -0.02& -0.019999& 1.33331\times {10}^{-6}& 0.00006667(.006)\\ -0.01& -0.0099998& 1.66666\times {10}^{-7}& 0.000016667\\ 0& 0& 0& 0\\ 0.01& 0.0099998& -1.66666\times {10}^{-7}& 0.000016667\\ 0.02& 0.019999& -1.33331\times {10}^{-6}& 0.00006667\\ 0.03& 0.029996& -4.4998\times {10}^{-6}& 0.00015\end{array}$

Now using graph to find value of $\left|\theta \right|$ for which error in approximating $\sin \theta \text{ by }\theta$ less than 1% of $\sin \theta$ .

See the following graph:

  

Here, for $\left|\theta \right|<0.2441$ error in approximating $\sin \theta \text{ by }\theta$ less than 1% of $\sin \theta$ .