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The problem revolves around an object moving around the unit circle, where the coordinates of the point \( P(t) \) are defined as \( x = \cos(t) \) and \( y = \sin(t) \), with \( 0 < t < \frac{\pi}{2} \). You are asked to calculate:

### Part (a): Slope of the tangent line at \( P(t) \)

The parametric equations for the unit circle are given by:
\[
x(t) = \cos(t), \quad y(t) = \sin(t)
\]
To find the slope of the tangent line at \( P(t) \), we differentiate \( y \) with respect to \( x \) using the chain rule:
\[
\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{\cos(t)}{-\sin(t)} = -\cot(t)
\]
Thus, the slope of the tangent line at \( P(t) \) is:
\[
\boxed{-\cot(t)}
\]

### Part (b): Equation of the tangent line

We will use the point-slope form of the equation of a line:
\[
y - y_1 = m(x - x_1)
\]
Here, \( m = -\cot(t) \) (the slope from part (a)), and the point \( P(t) \) has coordinates \( (\cos(t), \sin(t)) \).

So, the equation of the tangent line becomes:
\[
y - \sin(t) = -\cot(t)(x - \cos(t))
\]
This simplifies to the final equation of the tangent line.

Let me know if you would like more details or clarifications!

Here are 5 related questions:
1. How does the slope change as \( t \to 0 \) and \( t \to \frac{\pi}{2} \)?
2. What are the x- and y-intercepts of the tangent line at different points?
3. Can you derive the tangent line equation for a general parametric curve?
4. How would the equation change for a circle of radius \( r \)?
5. What happens to the tangent line when \( t = \frac{\pi}{4} \)?

**Tip:** To find the intercepts of a line, set \( x = 0 \) for the y-intercept and \( y = 0 \) for the x-intercept.

An object is moving around the unit circle so its x and y coordinates change with time as x = cos(t) and y = sin(t). Assume 0 < t < π/2. At a given time t, the tangent line to the unit circle at the position P(t) will determine a right triangle in the first quadrant where two corners are at the x and y intercepts, and the third one is at the origin. (Note: Both the point and slope of the tangent line you need for the line equation will depend on t.)

This problem explores an object moving along the unit circle, with coordinates x = cos(t) and y = sin(t). It involves finding the slope and equation of the tangent line at a given point P(t), using trigonometry and calculus. Perfect for advanced high school students and first-year college calculus courses.

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