Using Differentials to Approximate Sin 29

In this tutorial we shall look at the differentials of independent and dependent variables. Some applications of differentials will be discussed.

Use differentials to approximate the value of $$\sin {29^ \circ }$$

The nearest number to 29 whose sine value can be taken is 30, so let us consider that $$x = {30^ \circ }$$ and $$\delta x = dx = – {1^ \circ }$$.

Now consider
\[y = \sin x\,\,\,\,\,\,{\text{ – – – }}\left( {\text{i}} \right)\]
\[\begin{gathered} y + \delta y = \sin \left( {x + \delta x} \right) \\ \Rightarrow \sin \left( {x + \delta x} \right) = y + \delta y\,\,\,\,{\text{ – – – }}\left( {{\text{ii}}} \right) \\ \end{gathered} \]

Since $$y = \sin x,\,\,\delta y \approx dy$$, putting these values in equation (ii), we have
\[\sin \left( {x + \delta x} \right) \approx \sin x + \delta y\,\,\,\,{\text{ – – – }}\left( {{\text{iii}}} \right)\]

Taking the differential of equation (i), we have
\[dy = d\left( {\sin x} \right) = \cos dx\]

Putting this value in equation (ii), we have
\[\begin{gathered} \sin \left( {x + \delta x} \right) \approx \sin x + \cos xdx \\ \Rightarrow \sin \left( {{{30}^ \circ } – {1^ \circ }} \right) \approx \sin {30^ \circ } + \cos {30^ \circ }\left( { – {1^ \circ }} \right)\,\,\,\,\,\because x = {30^ \circ },\,\,dx = – {1^ \circ } = \delta x \\ \Rightarrow \sin {29^ \circ } \approx 0.5 + 0.866\left( { – 0.0174} \right)\,\,\,\,\,\because {1^ \circ } = 0.0174 \\ \Rightarrow \sin {29^ \circ } \approx 0.5 – \left( {0.866} \right)\left( {0.0174} \right) \\ \Rightarrow \sin {29^ \circ } \approx 0.5 – 0.0151 = 0.4849 \\ \end{gathered} \]