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