Question

Use a linear approximation (or differentials) to estimate the given number.36. \(\cos {29^ \circ }\)

Short answer

The required value is: \(\cos {29^ \circ } = 0.875\)

Step-by-step solution

Differentials

Thedifferentials can be taken as the substitute method of theLinear Approximation in calculus to approximate the value of any function.

Calculating valuesusing differential operators:

The givennumber is:

\(f\left( x \right)= \cos {29^ \circ } = \cos x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,..........\left( {{\rm{say}}} \right)\)

Now, solving the linear approximation at\(a = {30^ \circ }\):

\(\begin{aligned}{c}dy&= \left\{ {f'\left( a \right)} \right\}dx\\dy&= \left\{ {\left( { - \sin a} \right)\left| {_{a= {{30}^ \circ }}} \right.} \right\}\left( { - {1^ \circ }} \right)\\&= \left( { - \frac{1}{2}} \right)\left( { - \frac{\pi }{{180}}} \right)\\&= \frac{\pi }{{360}}\end{aligned}\)

So, for\(x = {29^ \circ }\):

\(\begin{aligned}{c}\cos {29^ \circ }&= f\left( {{{30}^ \circ }} \right) + \frac{\pi }{{360}}\\&= \frac{{\sqrt 3 }}{2} + \frac{\pi }{{360}}\\&= 0.875\end{aligned}\)

Hence, the required value is:

\(\cos {29^ \circ } = 0.875\)