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: