Graph the curve with parametric equations x=(1+cos16 t) cost y=(1+cos16 t) sint

step 1 So for this problem, we want to discuss how we can graph functions three -dimensionally. So what

Graph the curve with parametric equations x=(1+cos16 t)...

Graph the curve with parametric equations

$$
\begin{aligned}
& x=(1+\cos 16 t) \cos t \\
& y=(1+\cos 16 t) \sin t \\
& z=1+\cos 16 t
\end{aligned}
$$

Explain the appearance of the graph by showing that it lies on a cone.

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Key Concepts

Parametric Equations

Parametric equations describe curves in terms of a parameter (in this case, t) by expressing the coordinates (x, y, z) as functions of t. This approach allows one to describe complex, possibly non-function curves in space and to easily capture the dynamics or periodicity inherent in the system.

Elimination of the Parameter

Eliminating the parameter involves combining the equations to remove the variable t, resulting in an implicit equation that relates x, y, and z directly. In this example, squaring and adding the x and y components, and comparing with z, reveals a geometric constraint that the curve satisfies.

Trigonometric Identities and Polar Coordinates

The expressions for x and y involve cosine and sine functions, which naturally lend themselves to polar coordinates. Recognizing that cos²t + sin²t = 1 allows one to combine x and y into a single radial term, simplifying the process of eliminating the parameter and exposing the inherent symmetry of the curve.

Conic Surfaces (Cones)

A cone in three dimensions is defined by a quadratic relation between the spatial coordinates, commonly expressed as z² = x² + y² (possibly up to a scaling factor). By manipulating the parametric equations in the problem, one can show that the curve lies on a conical surface, explaining its appearance and spatial constraints.

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