# Using Differentials to Approximate Csc 61

In this tutorial we shall develop the differentials to approximate the value of $\csc {61^ \circ }$.

The nearest number to 61 degrees whose cosecant value can be taken is 60 degrees, so let us consider that $x = {60^ \circ }$ and $dx = {1^ \circ } = \frac{\pi }{{180}} = 0.0174$.

Now consider
$y = \csc x\,\,\,\,\,{\text{ – – – }}\left( {\text{i}} \right)$

Differentiating equation (i) with respect to $x$, we have
$\begin{gathered} \frac{{dy}}{{dx}} = \frac{d}{{dx}}\csc x \\ \Rightarrow \frac{{dy}}{{dx}} = – \csc x\cot x\,\,\,\,\,\,{\text{ – – – }}\left( {{\text{ii}}} \right) \\ \end{gathered}$

Taking the differential of equation (ii), we get
$\Rightarrow dy = – \csc x\cot xdx$

Using the values $x = {60^ \circ }$ and $dx = 0.0174$, we have
$\begin{gathered} dy = – \csc {60^ \circ }\cot {60^ \circ }\left( {0.0174} \right) \\ \Rightarrow dy = – \left( {\frac{2}{{\sqrt 3 }}} \right)\left( {\frac{1}{{\sqrt 3 }}} \right)\left( {0.0174} \right) = – 0.0114 \\ \end{gathered}$

Now
$\begin{gathered} \csc {61^ \circ } = y + dy \\ \Rightarrow \csc {61^ \circ } = \csc x + dy \\ \Rightarrow \csc {61^ \circ } = \csc {60^ \circ } – 0.0114 \\ \Rightarrow \csc {61^ \circ } = \frac{2}{{\sqrt 3 }} – 0.0114 \\ \Rightarrow \csc {61^ \circ } = 1.142 \\ \end{gathered}$