91Ó°ÊÓ

When dealing with parametric equations, calculating the derivative requires a slightly different approach compared to standard Cartesian equations. Here, derivatives help determine how the curve behaves as the parameter, in this case \( \theta \), changes.
To find the derivative \( \frac{dy}{dx} \) when \( y = 3 \sin \theta \) and \( x = 3 \cos \theta \), we first find the derivatives with respect to \( \theta \):

• \( \frac{dy}{d\theta} = 3 \cos \theta \)
• \( \frac{dx}{d\theta} = -3 \sin \theta \)

The next step is to divide the two derivatives to find \( \frac{dy}{dx} \):\[ \frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} = \frac{3 \cos \theta}{-3 \sin \theta} = -\cot \theta \]
This resulting form, \(-\cot \theta\), is crucial for finding points of tangency on the curve because it describes the slope of the tangent line at any point determined by \( \theta \).

A tangent to a curve is a line that just "touches" the curve at a particular point. Points of tangency are important in understanding the geometry and slopes of a curve. In this exercise, we are particularly focused on horizontal and vertical tangents.

• **Horizontal Tangency:** For horizontal tangency, the slope \( \frac{dy}{dx} \) of the tangent line should be 0. For the derived equation \(-\cot \theta = 0\), this occurs when \( \theta = \frac{\pi}{2} + n\pi \), where \( n \) is an integer. By substituting these values of \( \theta \) into the original parametric equations, you find the points on the curve where horizontal tangents exist.
• **Vertical Tangency:** A vertical tangency occurs where \(\frac{dy}{dx} \) becomes undefined, which for \(-\cot \theta \) is when \( \sin \theta = 0 \), or \( \theta = n\pi \). Again, inserting these \( \theta \) values into the equations helps us to locate where on the curve the vertical tangents appear.

These concepts are important for graph analysis as they can help you identify where curves change directions or exhibit asymptotic behavior.

Trigonometric functions play a vital role in parametric equations, especially when representing curves like circles or ellipses. In the given equations:

• \( x = 3 \cos \theta \)
• \( y = 3 \sin \theta \)

these represent a circle in the parametric form because of the intrinsic properties of \( \sin \theta \) and \( \cos \theta \).
The trigonometric identity \( \cos^2 \theta + \sin^2 \theta = 1 \) highlights that any parametric equation of this format maps onto a circle. Here, \( 3 \) acts as a radius, depicting a path where a point moves along the circle with changing \( \theta \).
Understanding these functions is essential because they not only define the shape and nature of the curve but also come into play when calculating derivatives and determining tangency points. They illustrate how the cycle of \( \theta \) affects \( x \) and \( y \) positions, allowing us to find precise points for analysis.