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Let's work through this step by step. The problem gives the parametric equations for the astroid:

\[
x = a \cos^3 \theta \quad \text{and} \quad y = a \sin^3 \theta
\]

### Step 1: Find the Slope of the Tangent Line

The slope of the tangent line for a parametric curve is given by:

\[
\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}
\]

We need to differentiate both \(x\) and \(y\) with respect to \(\theta\):

\[
\frac{dx}{d\theta} = \frac{d}{d\theta}(a \cos^3 \theta) = 3a \cos^2 \theta (-\sin \theta) = -3a \cos^2 \theta \sin \theta
\]

\[
\frac{dy}{d\theta} = \frac{d}{d\theta}(a \sin^3 \theta) = 3a \sin^2 \theta \cos \theta
\]

Now, the slope of the tangent line is:

\[
\frac{dy}{dx} = \frac{3a \sin^2 \theta \cos \theta}{-3a \cos^2 \theta \sin \theta} = -\frac{\sin \theta}{\cos \theta} = -\tan \theta
\]

So the slope of the tangent line is \( -\tan \theta \).

### Step 2: Find When the Tangent is Horizontal

A horizontal tangent occurs when \( \frac{dy}{dx} = 0 \), which happens when \( \frac{dy}{d\theta} = 0 \).

Since:

\[
\frac{dy}{d\theta} = 3a \sin^2 \theta \cos \theta
\]

For this to be zero, we need:

\[
\sin^2 \theta \cos \theta = 0
\]

This gives us the following cases:

1. \( \sin \theta = 0 \), which happens at \( \theta = 0 \) and \( \theta = \pi \).
2. \( \cos \theta = 0 \), but this won't satisfy the equation since \( \sin^2 \theta \) will be nonzero.

Thus, the horizontal tangents occur at \( \theta = 0 \) and \( \theta = \pi \).

### Step 3: Find When the Tangent is Vertical

A vertical tangent occurs when \( \frac{dx}{d\theta} = 0 \).

Since:

\[
\frac{dx}{d\theta} = -3a \cos^2 \theta \sin \theta
\]

For this to be zero, we need:

\[
\cos^2 \theta \sin \theta = 0
\]

This gives us the following cases:

1. \( \sin \theta = 0 \), but this won't satisfy the equation since \( \cos^2 \theta \) will be nonzero.
2. \( \cos \theta = 0 \), which happens at \( \theta = \frac{\pi}{2} \) and \( \theta = \frac{3\pi}{2} \).

Thus, the vertical tangents occur at \( \theta = \frac{\pi}{2} \) and \( \theta = \frac{3\pi}{2} \).

### Summary of Results:
- **Slope of the tangent line**: \( -\tan \theta \)
- **Horizontal tangents**: \( \theta = 0 \) and \( \theta = \pi \)
- **Vertical tangents**: \( \theta = \frac{\pi}{2} \) and \( \theta = \frac{3\pi}{2} \)

Would you like more details on any specific part of this? Let me know!

Here are 5 related questions you can explore:
1. How do we interpret the shape of an astroid based on the parametric equations?
2. How would the parametric equations change if the astroid had a different size?
3. What is the geometric significance of vertical and horizontal tangents?
4. How do we derive the equation for the tangent line in Cartesian form using parametric derivatives?
5. Can we use this method for other curves defined by parametric equations?

**Tip**: When dealing with parametric curves, always check where the derivatives of \(x\) or \(y\) with respect to the parameter (here, \(\theta\)) go to zero to find vertical or horizontal tangents.

Find the slope of the tangent line to the astroid in terms of θ. (a) At what points is the tangent horizontal? (b) At what points is the tangent vertical? Given the parametric equations: x = a cos^3 θ and y = a sin^3 θ

Learn how to find the slope of the tangent line to an astroid curve defined by x = a cos^3 θ and y = a sin^3 θ using parametric derivatives. Explore conditions for horizontal and vertical tangents, with a step-by-step explanation. Suitable for Grade 12 or college-level calculus.

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