Here is a plot of the triangle for different values of $\theta$, showing how the triangle changes as $\theta$ varies. The red point represents the fixed vertex $A(3, 4)$, while the other two vertices, $B(5\cos\theta, 5\sin\theta)$ and $C(5\sin\theta, -5\cos\theta)$, move according to $\theta$.

Next, to find the locus of the orthocenter, we need to calculate the intersection point of the altitudes for each value of $\theta$. Would you like to proceed with the algebraic solution to derive the equation of the orthocenter's locus?

Here are five questions to consider:

1. How is the slope of a line between two points calculated?
2. What is the relationship between the slopes of perpendicular lines?
3. How do you find the equation of a line given a point and a slope?
4. What is an orthocenter in a triangle, and how is it related to altitudes?
5. How does varying $\theta$ affect the positions of the triangle's vertices?

Tip: The orthocenter is always the point where the three altitudes of a triangle intersect.