Match the parametric equations with the graphs (labeled I–VI). Give reasons for your choices. x=

step 1 Given x equals cosine t, y equals sine t and z equals cosine of two t, here x is equal to cosine

Match the parametric equations with the graphs (labeled...

Key Concepts

Trigonometric Identities and Double-Angle Formulas

The expression z = cos 2t makes use of the double-angle identity in trigonometry. Recognizing the relationship between cos 2t and the functions of cos t or sin t (such as cos 2t = 2cos² t – 1) provides insight into how the z-coordinate oscillates relative to the circular motion in the xy-plane, linking the behavior of the vertical component to the position on the circle.

Projection and Spatial Visualization of Curves

This involves understanding how a 3D curve can be viewed from various angles by examining its projections onto coordinate planes. For example, while the xy-projection is a circle due to x = cos t and y = sin t, the variation of z with t (given by z = cos 2t) shows periodic behavior that adds depth to the curve’s structure. Visualizing these projections is key to matching the parametric equations with the correct graph among several provided options.

Parametric Equations in Three Dimensions

This concept involves representing curves in 3D space by expressing each coordinate (x, y, z) as a function of a common parameter, typically t. This method enables the analysis of the trajectory of points in space, facilitating understanding of spatial relationships and the curve's behavior when projected onto various coordinate planes.

Circular Motion in the XY-Plane

The equations x = cos t and y = sin t describe the standard parameterization of a circle in the xy-plane. This serves as a fundamental example in understanding how trigonometric functions can be used to represent closed curves, and it indicates that the projection of the spatial curve onto the xy-plane is a circle.

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