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Let's break down the problem and work through it step by step.

### Given Information:
- Function \( f(x) = \sin(x) \).
- Point \( (7, \sin(7)) \) which is approximately \( (7, 0.6570) \).
- Tangent line equation \( T(x) = 0.7539x - 4.6203 \).

### Problem Outline:
You are asked to:
1. Use the linear approximation (tangent line) to complete the table, where you have \( f(x) \) and \( T(x) \) values for different \( x \) values (around \( x = 7 \)).
2. Round your answers to four decimal places.

### Tangent Line \( T(x) \):
The equation of the tangent line at \( x = 7 \) is given as:
\[
T(x) = 0.7539x - 4.6203
\]
This will be used to approximate values of \( f(x) \) near \( x = 7 \).

### Step-by-Step Approach:

1. **Calculate \( f(x) = \sin(x) \)** at \( x = 6.9, 6.99, 7.01, 7.1 \). 

2. **Use the tangent line \( T(x) = 0.7539x - 4.6203 \)** to estimate \( f(x) \) at the same values of \( x \) by plugging the values into the linear approximation equation.

Let's calculate these values:

### 1. \( f(x) = \sin(x) \):

- \( f(6.9) = \sin(6.9) = -0.5225 \)
- \( f(6.99) = \sin(6.99) = 0.5867 \)
- \( f(7.01) = \sin(7.01) = 0.7269 \)
- \( f(7.1) = \sin(7.1) = 0.7626 \)

### 2. \( T(x) = 0.7539x - 4.6203 \):

- \( T(6.9) = 0.7539(6.9) - 4.6203 = 0.5656 \)
- \( T(6.99) = 0.7539(6.99) - 4.6203 = 0.6576 \)
- \( T(7.01) = 0.7539(7.01) - 4.6203 = 0.6582 \)
- \( T(7.1) = 0.7539(7.1) - 4.6203 = 0.6585 \)

### Completed Table:

\[
\begin{array}{|c|c|c|}
\hline
x & f(x) & T(x) \\
\hline
6.9 & -0.5225 & 0.5656 \\
6.99 & 0.5867 & 0.6576 \\
7.00 & 0.6570 & 0.6570 \\
7.01 & 0.7269 & 0.6582 \\
7.1 & 0.7626 & 0.6585 \\
\hline
\end{array}
\]

If you have any further questions or want more details, feel free to ask!

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### Related Questions:
1. How is the equation of the tangent line derived?
2. What is the significance of using a tangent line as a linear approximation?
3. Why is the sine function used for the example problem?
4. How can linear approximation be applied to non-linear functions?
5. What is the general formula for the equation of a tangent line at a point?

### Tip:
Linear approximations work well for values close to the point of tangency but become less accurate as you move further from that point.

Find the equation of the tangent line T to the graph of f at the given point. Use this linear approximation to complete the table. (Round your answers to four decimal places.) f(x) = sin x, (7, sin 7)

Learn how to find the tangent line to the graph of f(x) = sin(x) at the given point (7, sin(7)) and use this linear approximation to complete a table of values. This example demonstrates key concepts in calculus, including linear approximation and trigonometric functions. Suitable for students in Grades 10-12.

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