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Chapter 4: Problem 30

Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error. $$\cos 31^{\circ}$$

Short Answer

Expert verified
Question: Estimate the value of \(\cos 31^{\circ}\) using linear approximations. Answer: The linear approximation of \(\cos 31^{\circ}\) is approximately \(\frac{\sqrt{3}}{2} - \frac{\pi}{360}\).

Step by step solution

01

Convert to radians

First, convert the given angle from degrees to radians, as trigonometric functions are defined in terms of radians. $$\theta = 31^{\circ} \times \frac{\pi}{180^{\circ}} = \frac{31\pi}{180}\,rad$$
02

Choose a suitable value for approximation

To get a small error, choose a nearby angle value where the cosine function can be easily evaluated. We'll use \(30^{\circ}\) or \(\frac{\pi}{6}\) radians as our point of approximation. $$a = \frac{\pi}{6}\,rad$$
03

Calculate the difference

Calculate the difference between the given angle and the point of approximation: $$\Delta\theta = \theta - a = \frac{31\pi}{180} - \frac{\pi}{6} = \frac{\pi}{180}\,rad$$
04

Use the first-degree Taylor series expansion

Use the first-degree Taylor series expansion of the cosine function around the point \(a\): $$\cos (\theta) \approx \cos(a) - \sin(a)(\theta - a)$$
05

Substitute the values

Substitute the values for \(a\), the known cosine and sine values at \(\frac{\pi}{6}\), and \(\Delta \theta\) : $$\cos \left(\frac{31\pi}{180}\right) \approx \cos\left(\frac{\pi}{6}\right) - \sin\left(\frac{\pi}{6}\right) \cdot \frac{\pi}{180}$$
06

Evaluate the expression

Evaluate the expression to get the approximate value for \(\cos 31^{\circ}\): $$\cos\left(\frac{31\pi}{180}\right) \approx \frac{\sqrt{3}}{2} - \frac{1}{2}\cdot\frac{\pi}{180}$$ So, the linear approximation of \(\cos 31^{\circ}\) is approximately \(\frac{\sqrt{3}}{2} - \frac{\pi}{360}\).

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