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To calculate \( \cos 25^\circ \) and \( \cos 73^\circ \) using Newton's interpolation formula, we'll proceed with the following steps:

### Newton’s Forward Difference Interpolation Formula:
The Newton forward difference interpolation formula is given as:

\[
f(x) = f(x_0) + \frac{(x - x_0)}{1!} \Delta f(x_0) + \frac{(x - x_0)(x - x_1)}{2!} \Delta^2 f(x_0) + \cdots
\]

Where:
- \( f(x) \) is the interpolated value of the function.
- \( x_0 \) is the first value of \( x \) in the table.
- \( \Delta f(x_0) \), \( \Delta^2 f(x_0) \), etc., are the forward differences.

We'll calculate both \( \cos 25^\circ \) and \( \cos 73^\circ \) by applying this method using the data provided in the table:

\[
\begin{array}{|c|c|}
\hline
x^\circ & \cos x \\
\hline
10 & 0.9848 \\
20 & 0.9397 \\
30 & 0.8660 \\
40 & 0.7660 \\
50 & 0.6428 \\
60 & 0.5000 \\
70 & 0.3420 \\
80 & 0.1737 \\
\hline
\end{array}
\]

Let me compute the forward differences and apply Newton's formula for both \( 25^\circ \) and \( 73^\circ \).The forward difference table is computed as shown below:

\[
\begin{array}{|c|c|c|c|c|c|c|c|c|}
\hline
x^\circ & \cos x & \Delta^1 & \Delta^2 & \Delta^3 & \Delta^4 & \Delta^5 & \Delta^6 & \Delta^7 \\
\hline
10 & 0.9848 & -0.0451 & -0.0286 & 0.0023 & 0.0008 & -0.0003 & 0.0006 & -0.0012 \\
20 & 0.9397 & -0.0737 & -0.0263 & 0.0031 & 0.0005 & 0.0003 & -0.0006 & 0.0000 \\
30 & 0.8660 & -0.1000 & -0.0232 & 0.0036 & 0.0008 & -0.0003 & 0.0000 & 0.0000 \\
40 & 0.7660 & -0.1232 & -0.0196 & 0.0044 & 0.0005 & 0.0000 & 0.0000 & 0.0000 \\
50 & 0.6428 & -0.1428 & -0.0152 & 0.0049 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\
60 & 0.5000 & -0.1580 & -0.0103 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\
70 & 0.3420 & -0.1683 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\
80 & 0.1737 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\
\hline
\end{array}
\]

Now, let's apply Newton's forward difference interpolation to estimate \( \cos 25^\circ \) and \( \cos 73^\circ \).

### Interpolation for \( \cos 25^\circ \):
We will

Calculate cos 25° and cos 73° using Newton's interpolation formula based on the given table of cos x values.

Learn how to calculate cos 25° and cos 73° using Newton’s Forward Difference Interpolation formula. This solution covers trigonometry concepts and the application of interpolation techniques to estimate values between given data points. Suitable for advanced high school and undergraduate students.

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