Problem 30 - Calculus Early Transcendentals

Question

Use linear approximations to estimate the following quantities. Choose a value of a that produces a small error. \(\cos 31^{\circ}\)

Step-by-Step Solution

Answer

Question: Estimate the value of cos(31°) using linear approximation. Answer: cos(31°) is approximately 0.8572 using linear approximation.

Step 1: Choose an appropriate value for a

Choose a value of \(a\) close to \(31^\circ\) whose cosine value is easily known. An appropriate choice is 30 degrees, since the cosine of \(30^\circ\) (or \(\frac{\pi}{6}\) radians) is well known. So, we have \(a = 30^\circ\).

Step 2: Convert degrees to radians

Convert \(31^\circ\) and \(30^\circ\) to radians, as trigonometric functions work with radians. We know that \(180^\circ = \pi\) radians, so: \(31^\circ = \frac{31\pi}{180}\) radians, \(30^\circ = \frac{30\pi}{180} = \frac{\pi}{6}\) radians.

Step 3: Find the derivative of the cosine function

The linear approximation is based on using the derivative of the function. In our case, the function is \(\cos x\). Let's find the derivative of \(\cos x\). We know that: \(\frac{d}{dx}(\cos x) = -\sin x\).

Step 4: Evaluate the derivative of the function at the chosen point

We need to find the value of the derivative at our chosen value of \(a = \frac{\pi}{6}\). So, evaluate \(-\sin a\) at \(a = \frac{\pi}{6}\): \(-\sin \left(\frac{\pi}{6}\right) = -\frac{1}{2}\).

Step 5: Apply the linear approximation formula

The linear approximation at a point \(a\) is given by: \(f(x) \approx f(a) + f'(a)(x-a)\) In our case: \(\cos x \approx \cos \left(\frac{\pi}{6}\right) - \frac{1}{2}\left(x-\frac{\pi}{6}\right)\)

Step 6: Substitute the value of x and calculate the estimate

Now, substitute the value of \(x = \frac{31\pi}{180}\) into the approximation and calculate the estimate value for \(\cos 31^\circ\): \(\cos \left(\frac{31\pi}{180}\right) \approx \cos \left(\frac{\pi}{6}\right) - \frac{1}{2}\left(\frac{31\pi}{180}-\frac{\pi}{6}\right)\) \(\cos 31^\circ \approx \frac{\sqrt{3}}{2} - \frac{1}{2}\left(\frac{\pi}{180}\right)\) Now, you can use a calculator to find the approximate value of \(\cos 31^\circ\) using the linear approximation: \(\cos 31^\circ \approx 0.8572\)

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, particularly when dealing with angles and periodic phenomena. These functions include sine, cosine, and tangent, among others. In our exercise, we focus on the cosine function, which relates the angle in a right triangle to the ratio of the adjacent side to the hypotenuse.

Cosine is periodic with a range of [-1, 1] and has several well-known values at standard angles. For instance, \(\cos 30^\circ = \frac{\sqrt{3}}{2}\), a value used extensively in trigonometry. These known values are crucial in making calculations like linear approximations simpler.

Understanding the basic properties of trigonometric functions helps in approximations and transforms problems involving angles into solvable equations with familiar number ranges.

• Key properties: Periodicity and symmetry.
• Commonly used angles: \(0^\circ, 30^\circ, 45^\circ, 60^\circ,\) and \(90^\circ\).

Derivatives

Derivatives measure how a function changes as its input changes and are a central concept in calculus. In our linear approximation task, the derivative of cosine, specifically, helps us understand how cosine values change near a specific point.

The derivative of the cosine function is \(\frac{d}{dx}(\cos x) = -\sin x\). This tells us the rate of change of cosine at any angle \(x\). When estimating values with linear approximation, knowing how quickly cosine changes gives us a way to predict its value near a point of interest.

• Role in Linear Approximation: Provides slope for tangent line approximation at \(a\).
• Essentials to remember: The derivative’s effect and how it influences approximation accuracy.

By calculating the derivative at a specific point (like \(a = 30^\circ\)), we use it to find how close our approximation is to the true value.

Degrees to Radians Conversion

In trigonometry, angles can be measured in degrees or radians, with radians often preferred in mathematical calculations. Converting degrees to radians involves a straightforward formula:

\[x^\circ = \frac{x\pi}{180}\]
This conversion is essential because functions like sine and cosine are typically more naturally engaged in radians.

In our exercise, converting \(31^\circ\) to radians gives us \(\frac{31\pi}{180}\), aligning it with the radian format of cosine functions.

• Importance: Allows proper use of trigonometric formulas and calculus applications.
• How to Convert: Multiply degree value by \(\frac{\pi}{180}\).

Familiarity with this conversion allows you to work seamlessly between degrees and radians, crucial for accurate trigonometric and calculus operations.