Question

use a linear approximation to estimate the following quantity. choose a value of a that produces a small error and does not require the use of a calculator cos 31° the linear approximation of cos 31° is (type an exact answer)

Explanation:

Step1: Convert degrees to radians

We know that \(x = 31^{\circ}=\frac{31\pi}{180}\) radians and choose \(a = 30^{\circ}=\frac{\pi}{6}\) radians. The function is \(f(x)=\cos x\), and \(f^{\prime}(x)=-\sin x\).

Step2: Calculate \(f(a)\) and \(f^{\prime}(a)\)

\(f(a)=\cos\frac{\pi}{6}=\frac{\sqrt{3}}{2}\), \(f^{\prime}(a)=-\sin\frac{\pi}{6}=-\frac{1}{2}\)

Step3: Use the linear - approximation formula

The linear - approximation formula is \(L(x)=f(a)+f^{\prime}(a)(x - a)\). Substitute \(x=\frac{31\pi}{180}\) and \(a = \frac{\pi}{6}\) (where \(x - a=\frac{31\pi}{180}-\frac{\pi}{6}=\frac{31\pi - 30\pi}{180}=\frac{\pi}{180}\))

\(L(x)=\frac{\sqrt{3}}{2}-\frac{1}{2}\times\frac{\pi}{180}=\frac{\sqrt{3}}{2}-\frac{\pi}{360}\)

Answer:

\(\frac{\sqrt{3}}{2}-\frac{\pi}{360}\)