Chapter 4 - Applications of the Derivative - 4.5 Linear Approximation and Differentials - 4.5 Exercises - Page 289: 30

$$a = 30^\circ ,{\text{ }}\cos \left( {31^\circ } \right) \approx 0.87475205$$

$$\eqalign{ & \cos \left( {31^\circ } \right) \cr & {\text{Using the function }}f\left( x \right) = \cos x,\left( {{\text{Use the example 3 as a guide}}} \right) \cr & f\left( x \right) = \cos x \cr & \cr & {\text{we must convert to radian measure}}{\text{, because the derivative }} \cr & {\text{formulas for trigonometric functions require angles in radians}}{\text{.}} \cr & 30^\circ + 1^\circ = 30^\circ \left( {\frac{\pi }{{180^\circ }}} \right){\text{rad}} + 1^\circ \left( {\frac{\pi }{{180^\circ }}} \right){\text{rad}} \cr & 30^\circ + 1^\circ = \frac{\pi }{6}{\text{rad}} + \frac{\pi }{{180}}{\text{rad}} = \frac{{31\pi }}{{180}}{\text{rad}} \cr & {\text{we have that }}31^\circ = \frac{\pi }{6}{\text{rad}} + \frac{\pi }{{180}}{\text{rad}},{\text{ let }}a = \frac{\pi }{6} \cr & \cr & {\text{* Evaluate }}f\left( {\frac{\pi }{6}} \right) \cr & f\left( {\frac{\pi }{6}} \right) = \cos \left( {\frac{\pi }{6}} \right) \cr & f\left( {\frac{\pi }{6}} \right) = \frac{{\sqrt 3 }}{2} \cr & \cr & {\text{Differentiating we obtain}} \cr & f'\left( x \right) = \sin x \cr & {\text{* Evaluate at }}a = \frac{\pi }{6},{\text{ }}f'\left( {\frac{\pi }{6}} \right) \cr & f'\left( {\frac{\pi }{6}} \right) = \sin \left( {\frac{\pi }{6}} \right) = \frac{1}{2} \cr & \cr & {\text{Using the definition of a linear approximation to }}f{\text{ at }}a{\text{ }}\left( {page{\text{ 282}}} \right) \cr & L\left( x \right) = f\left( a \right) + f'\left( a \right)\left( {x - a} \right),{\text{ for an interval }}I{\text{ containing }}a \cr & \cr & {\text{The linear approximation }}L\left( x \right){\text{ at }}a{\text{ is:}} \cr & L\left( x \right) = \frac{{\sqrt 3 }}{2} + \frac{1}{2}\left( {x - \frac{\pi }{6}} \right) \cr & \cr & {\text{Estimating the given value at }}f\left( {\frac{{31\pi }}{{180}}} \right) \cr & L\left( {\frac{{31\pi }}{{180}}} \right) = \frac{{\sqrt 3 }}{2} + \frac{1}{2}\left( {\frac{{31\pi }}{{180}} - \frac{\pi }{6}} \right) \cr & L\left( {\frac{{31\pi }}{{180}}} \right) = 0.87475205 \cr & \cr & {\text{Therefore}}{\text{,}} \cr & \cos \left( {31^\circ } \right) \approx L\left( {\frac{{31\pi }}{{180}}} \right) \approx 0.87475205 \cr & {\text{The result into a calculator is 0}}{\text{.857167307}} \cr} $$