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}