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] radian
Radian 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] radian
Substitute 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]
