Let's revisit the problem to get a more accurate estimate of the slope of the tangent line at $P(1, 0)$.

Analyzing the Secant Slopes

To estimate the slope of the tangent line at $x = 1$, we should observe the pattern of the secant slopes as $x$ approaches 1. We calculated the following slopes:

• As $x$ gets closer to 1 from the left (e.g., $x = 0.9$, $0.8$, etc.), the slopes tend to increase significantly.
• From the right (e.g., $x = 1.1$, $1.2$, etc.), we also see an increase but with opposite signs.

Key Points to Consider:

1. Values approaching from the left:

   • $x = 0.9$: Slope $\approx 9.8481$
   • $x = 0.8$: Slope $\approx 5.0000$
   • $x = 0.7$: Slope $\approx 0.0000$
   • $x = 0.6$: Slope $\approx 2.1651$

2. Values approaching from the right:

   • $x = 1.1$: Slope $\approx 7.5575$
   • $x = 1.2$: Slope $\approx -4.3301$
   • $x = 1.3$: Slope $\approx 2.2104$
   • $x = 1.4$: Slope $\approx 0.0000$

Slope Estimation

The tangent slope at $x = 1$ can be better estimated by focusing on the values near $x = 1$. Observing the pattern, it appears that as $x$ approaches 1 from both sides, the slopes converge towards a positive value.

However, since the slope isn't directly zero and shows oscillating behavior, we need to look at the average behavior near the point. Both the positive and negative large values suggest a non-zero slope.

Refined Estimate

Taking the average behavior into account (particularly near $x = 1$), the slope of the tangent line at $P(1, 0)$ should be a positive value that considers the increasing magnitude from both sides:

Given the oscillation pattern, the refined slope estimate would be closer to 3.00 rather than zero.

Would you like further clarification on this refinement, or is there another aspect you'd like to explore? Here are some related questions:

1. Why does the slope estimate shift from negative to positive as $x$ moves across 1?
2. How can we improve the accuracy of tangent slope estimates using secant slopes?
3. What does the non-zero slope indicate about the curve at $x = 1$?
4. Would a similar pattern emerge for other trigonometric functions at different points?
5. Can we determine the slope more precisely by using a limit approach rather than secant lines?

Tip: When estimating the slope of a tangent using secant lines, always consider the behavior of the slope on both sides of the point to get a more accurate estimate.