https://file.aiquickdraw.com/mathbot/file/1725890252605wa9ztjok.jpg

The problem asks to find the value of \( \sec(31^\circ) \) using numerical differentiation. The given data provides \( \tan(\theta) \) for values of \( \theta = 31^\circ, 32^\circ, 33^\circ, 34^\circ \).

The hint provided states that:
\[
\frac{d}{d\theta} (\tan(\theta)) = \sec^2(\theta)
\]
We can numerically differentiate \( \tan(\theta) \) to estimate \( \sec^2(31^\circ) \), and then take the square root to find \( \sec(31^\circ) \).

### Step-by-Step Solution

We use the finite difference method for numerical differentiation. Since we are asked for the value at \( 31^\circ \), a forward difference approximation is suitable:

\[
\frac{d}{d\theta} (\tan(\theta)) \approx \frac{\tan(32^\circ) - \tan(31^\circ)}{32^\circ - 31^\circ}
\]

Substitute the values from the table:

\[
\frac{d}{d\theta} (\tan(\theta)) \approx \frac{0.6249 - 0.6008}{1^\circ} = 0.0241
\]

This value represents \( \sec^2(31^\circ) \), so:

\[
\sec^2(31^\circ) \approx 0.0241
\]

To find \( \sec(31^\circ) \):

\[
\sec(31^\circ) = \sqrt{0.0241} \approx 0.155
\]

This is the numerical value of \( \sec(31^\circ) \) using numerical differentiation. 

Would you like more details or a more refined method for accuracy? Let me know!

Here are 5 relative questions for practice:
1. How would you compute the value of \( \sec(32^\circ) \) using the same method?
2. What is the central difference formula for numerical differentiation?
3. How does the finite difference method work for non-equidistant points?
4. How is the second derivative computed using numerical differentiation?
5. What is the significance of the hint \( \frac{d}{d\theta} (\tan(\theta)) = \sec^2(\theta) \)?

**Tip**: Always check the step size when using numerical differentiation to ensure accuracy!

From the values in the table given below, find the value of sec 31° using numerical differentiation

Learn how to find the value of sec(31°) using numerical differentiation from the given table of tan(θ) values. This involves the forward difference method and understanding trigonometric identities such as sec²(θ). Suitable for engineering students and higher-level math learners.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation