Chapter 1, Problem 53GP

(a)

To determine

The percent uncertainty in $\theta$ and $\sin \theta$ .

Answer to Problem 53GP

The percent uncertainty in $\theta$ is 3.33%.

The percent uncertainty in $\sin \theta$ is 3.25%.

Explanation of Solution

Given:

The angle $\theta =15°\pm 0.5°$

Formula used:

The expression for percent uncertainty in $\theta$ is,

  $\%\text{uncertainty in }\theta =\frac{\Delta \theta }{\theta }\times 100\%$

Here, $\Delta \theta$ is the uncertainty and $\theta$ is the measured value.

The expression for percent uncertainty in $\sin \theta$ is,

  $\%\text{uncertainty in sin}\theta =\frac{\Delta \sin \theta }{\sin \theta }\times 100\%$

Here, $\Delta \sin \theta$ is the uncertainty and $\sin \theta$ is the measured value.

Calculation:

The percent uncertainty in $\theta$ is,

  $\begin{array}{c}\%\text{uncertainty in }\theta =\frac{\Delta \theta }{\theta }\times 100\%\\ =\frac{15.5°-15°}{15°}\times 100\\ =3.33\%\end{array}$

The percent uncertainty in $\sin \theta$ is,

  $\begin{array}{c}\%\text{uncertainty in sin}\theta =\frac{\Delta \sin \theta }{\sin \theta }\times 100\%\\ =\frac{\sin 15.5°-\sin 15°}{\sin 15°}\times 100\\ =3.25\%\end{array}$

Conclusion:

Thus, the percent uncertainty in $\theta$ is 3.33% and the percent uncertainty in $\sin \theta$ is 3.25%.

(b)

To determine

The percent uncertainty in $\theta$ and $\sin \theta$ .

Answer to Problem 53GP

The percent uncertainty in $\theta$ is 0.67%.

The percent uncertainty in $\sin \theta$ is 0.23%.

Explanation of Solution

Given:

The angle $\theta =75°\pm 0.5°$

Formula used:

The expression for percent uncertainty in $\theta$ is,

  $\%\text{uncertainty in }\theta =\frac{\Delta \theta }{\theta }\times 100\%$

Here, $\Delta \theta$ is the uncertainty and $\theta$ is the measured value.

The expression for percent uncertainty in $\sin \theta$ is,

  $\%\text{uncertainty in sin}\theta =\frac{\Delta \sin \theta }{\sin \theta }\times 100\%$

Here, $\Delta \sin \theta$ is the uncertainty and $\sin \theta$ is the measured value.

Calculation:

The percent uncertainty in $\theta$ is,

  $\begin{array}{c}\%\text{uncertainty in }\theta =\frac{\Delta \theta }{\theta }\times 100\%\\ =\frac{75.5°-75°}{75°}\times 100\\ =0.67\%\end{array}$

The percent uncertainty in $\sin \theta$ is,

  $\begin{array}{c}\%\text{uncertainty in sin}\theta =\frac{\Delta \sin \theta }{\sin \theta }\times 100\%\\ =\frac{\sin 75.5°-\sin 75°}{\sin 75°}\times 100\\ =0.23\%\end{array}$

Conclusion:

Thus, the percent uncertainty in $\theta$ is 0.67% and the percent uncertainty in $\sin \theta$ is 0.23%.