To find the points at which the brachistochrone has a horizontal tangent, we need to determine where the slope of the curve is zero. The slope of the curve is given by the derivative of $y(t)$ with respect to $x(t)$.

Given the parametric equations for the brachistochrone:

• $x(t) = t - \sin(t)$
• $y(t) = \cos(t) - 1$

We can find the derivative of $y$ with respect to $x$ using the chain rule:

$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$

Step 1: Compute the derivatives of $x(t)$ and $y(t)$

1. $\frac{dx}{dt} = 1 - \cos(t)$
2. $\frac{dy}{dt} = -\sin(t)$

Step 2: Set the slope equal to zero

To find where the tangent is horizontal, we need $\frac{dy}{dx} = 0$, which means: $\frac{dy}{dt} = 0$

Since $\frac{dy}{dx} = \frac{-\sin(t)}{1 - \cos(t)}$, the horizontal tangent occurs when $\frac{dy}{dt} = -\sin(t) = 0$, so:

$\sin(t) = 0$

Step 3: Solve for $t$

The sine function is zero at integer multiples of $\pi$, so: $t = n\pi \quad \text{for} \quad n = 0, 1, 2, \dots$

Step 4: Find the corresponding points

Substitute these values of $t$ back into the parametric equations to find the coordinates where the brachistochrone has a horizontal tangent.

1. For $t = 0$: $x(0) = 0 - \sin(0) = 0$ $y(0) = \cos(0) - 1 = 1 - 1 = 0$ So, the first point is $(0, 0)$.

2. For $t = \pi$: $x(\pi) = \pi - \sin(\pi) = \pi$ $y(\pi) = \cos(\pi) - 1 = -1 - 1 = -2$ So, the second point is $(\pi, -2)$.

Thus, the brachistochrone has horizontal tangents at the points $(0, 0)$ and $(\pi, -2)$.

Would you like more details or have any questions about this solution?

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Related questions:

1. How is the brachistochrone problem related to the concept of the cycloid?
2. What is the derivation of the parametric equations for a cycloid?
3. How does the brachistochrone curve differ from a straight line in terms of time of descent?
4. What is the history behind the brachistochrone problem in physics and mathematics?
5. How can we use the calculus of variations to derive the brachistochrone curve?

Tip: The brachistochrone problem is an excellent example of how physics and mathematics intersect to solve real-world optimization problems.