Use a linear approximation to estimate the following quantity. Choose a value of a to produce a small error. cos 24 degree What is the value found using the linear approximation? Convert any degree measures to radians. cos 24 degree [ ](Type an exact answer, using pi as needed.)

Accepted Solution

Answer:[tex] cos 24^{\circ}\approx 0.9184[/tex]Step-by-step explanation:We have to find the value of [tex]cos 24^{\circ}[/tex] using linear approximation.Let [tex]f(x)= cos x[/tex]and [tex]a=\frac{\pi}{6}=30^{\circ}[/tex]Differentiate w.r.t. x[tex]f'(x)=-sinx[/tex]Substitute the value of a then , we get [tex]f(\frac{\pi}{6})=cos\frac{\pi}{6}=\frac{\sqrt3}{2}[/tex][tex]f'(\frac{\pi}{3})=-sin\frac{\pi}{6}=-\frac{1}{2}[/tex]Linear approximation near a is given by the  formula:[tex]L(x)=f(a)+f'(a)(x-a)[/tex]We  have 24 degree which is near to 30 degree Therefore, we have a=30 degree=[tex]\frac{\pi}{6}[/tex] radianRadian measure=[tex]\frac{\pi}{180}\times degree\;measure[/tex]Convert 24 degree in to radian measure by above formula [tex]\frac{\pi}{180}\times 24=\frac{2\pi}{15}[/tex] radianSubstitute the values in the given formula[tex]L(\frac{2\pi}{15})\approx\frac{\sqrt3}{2}-\frac{1}{2}(\frac{2\pi}{15}-\frac{\pi}{6})[/tex][tex]cos(\frac{2\pi}{15})\approx\frac{\sqrt3}{2}-\frac{1}{2}\times (-\frac{\pi}{30})[/tex][tex]cos(\frac{2\pi}{15})\approx\frac{\sqrt3}{2}+\frac{\pi}{60}[/tex][tex] cos 24^{\circ}\approx 0.9184[/tex]