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To solve the problem of finding \(\sec 31^\circ\) using numerical differentiation, we will use the hint provided, which states that:

\[
\frac{d}{d\theta} (\tan \theta) = \sec^2 \theta
\]

Given that:

\[
\sec^2(31^\circ) = \frac{d}{d\theta} (\tan \theta) \bigg|_{\theta = 31^\circ}
\]

We will estimate the derivative \(\frac{d}{d\theta} (\tan \theta)\) at \(\theta = 31^\circ\) using the values from the table by applying a central difference formula:

\[
\frac{d (\tan \theta)}{d\theta} \bigg|_{\theta = 31^\circ} \approx \frac{\tan(32^\circ) - \tan(30^\circ)}{32^\circ - 30^\circ}
\]

However, the table doesn't provide \(\tan(30^\circ)\), so we instead use the forward difference formula:

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

Substituting the values:

\[
\frac{d (\tan \theta)}{d\theta} \bigg|_{\theta = 31^\circ} \approx \frac{0.6249 - 0.6008}{1^\circ} = 0.0241
\]

Thus,

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

Taking the square root to find \(\sec(31^\circ)\):

\[
\sec(31^\circ) \approx \sqrt{0.0241} \approx 0.1552
\]

However, this seems incorrect for a value of \(\sec 31^\circ\). Let me correct this by using more appropriate central difference estimation.

Let's reattempt using a central difference formula:

\[
\frac{d (\tan \theta)}{d\theta} \bigg|_{\theta = 32^\circ} \approx \frac{\tan(33^\circ) - \tan(31^\circ)}{33^\circ - 31^\circ} = \frac{0.6494 - 0.6008}{2^\circ} = \frac{0.0486}{2^\circ} = 0.0243
\]

So:

\[
\sec^2(32^\circ) \approx 0.0243
\]

And then:

\[
\sec(32^\circ) \approx \sqrt{0.0243} = 0.1559
\]

There seems to be an error; the correct process should involve applying the derivative correctly. The correct method involves recognizing that \(\frac{d (\tan \theta)}{d\theta} \bigg|_{\theta = 31^\circ}\) represents a much higher number, because:

\[
\sec^2(31^\circ) \approx \text{(Large number near 1)}
\]

Therefore, estimating \(\sec^2\) requires evaluating sec based on a small derivative. The precise solution, if done analytically, would yield \(\sec 31^\circ\) closer to 1. The hint must be used with accurate trigonometric estimates.

Do you want me to continue with more detailed steps or questions on this?

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

Solve the problem of finding sec 31° using numerical differentiation by applying central and forward difference formulas. Learn key trigonometric differentiation techniques and formulas, including the derivative of tan(θ) and secant functions.

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