Use Differentials to Approximate Cos 44

In this tutorial we shall be concerned with the Use differentials to approximate the value of \cos {44^ \circ }.

The nearest number to 44 whose cosine value can be taken is 45, so let us consider that x = {45^ \circ } and \delta x = dx = - {1^ \circ }.

Now consider

y = \cos x\,\,\,\,\,\,{\text{ - - - }}\left( {\text{i}} \right)


\begin{gathered} y + \delta y = \cos \left( {x + \delta x} \right) \\ \Rightarrow \cos \left( {x + \delta x} \right) = y + \delta y\,\,\,\,{\text{ - - - }}\left( {{\text{ii}}} \right) \\ \end{gathered}

Since y = \cos x,\,\,\delta y \approx dy, so putting these values in equation (ii), we have

\cos \left( {x + \delta x} \right) \approx \cos x + \delta y\,\,\,\,{\text{ - - - }}\left( {{\text{iii}}} \right)

Taking differential of equation (i), we have

dy = d\left( {\cos x} \right) = - \sin dx

Putting this value in equation (ii), we have

\begin{gathered} \cos \left( {x + \delta x} \right) \approx \cos x + \left( { - \sin xdx} \right) \\ \Rightarrow \cos \left( {x + \delta x} \right) \approx \cos x - \sin xdx \\ \Rightarrow \cos \left( {{{45}^ \circ } - {1^ \circ }} \right) \approx \cos {45^ \circ } - \sin {45^ \circ }\left( { - {1^ \circ }} \right)\,\,\,\,\,\because x = {45^ \circ },\,\,dx = - {1^ \circ } = \delta x \\ \Rightarrow \cos {44^ \circ } \approx 0.707 - 0.707\left( { - 0.0174} \right)\,\,\,\,\,\because {1^ \circ } = 0.0174 \\ \Rightarrow \cos {44^ \circ } \approx 0.707 + 0.0123 \\ \Rightarrow \cos {44^ \circ } \approx 0.719 \\ \end{gathered}