# Using Differentials to Approximate Cos 44

In this tutorial we shall look at the use of 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 $y = \cos x,\,\,\delta y \approx dy$, 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 the differential of equation (i), we have
$dy = d\left( {\cos x} \right) = – \sin dx$

Putting this value in equation (ii), we have
$\begin{gathered} \cos \left( {x + \delta x} \right) \approx \cos x + \left( { – \sin xdx} \right) \\ \Rightarrow \cos \left( {x + \delta x} \right) \approx \cos x – \sin xdx \\ \Rightarrow \cos \left( {{{45}^ \circ } – {1^ \circ }} \right) \approx \cos {45^ \circ } – \sin {45^ \circ }\left( { – {1^ \circ }} \right)\,\,\,\,\,\because x = {45^ \circ },\,\,dx = – {1^ \circ } = \delta x \\ \Rightarrow \cos {44^ \circ } \approx 0.707 – 0.707\left( { – 0.0174} \right)\,\,\,\,\,\because {1^ \circ } = 0.0174 \\ \Rightarrow \cos {44^ \circ } \approx 0.707 + 0.0123 \\ \Rightarrow \cos {44^ \circ } \approx 0.719 \\ \end{gathered}$