Use Differentials to Approximate Csc 61

In this tutorial we shall develop the differentials approximate the value of \csc  {61^ \circ }.
The nearest number to 61 degree whose cosecant value can be taken is 60 degree, 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)


Differentiate 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


Putting 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}

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