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Parametric equations are a powerful tool for describing curves in mathematics. Unlike standard Cartesian equations that directly relate x and y, parametric equations describe a path by expressing x and y (and sometimes z for 3D curves) as functions of an independent parameter, usually denoted as t.

In our exercise, the parametric equations are given as: \[\mathbf{r}(t) = \langle 10 \cos t, 2 \sin t, 1 \rangle\]This equation represents a three-dimensional curve where each component of the vector \( \mathbf{r}(t) \) is a function of t. For the x-component, we have \( x(t) = 10 \cos t \), for y, \( y(t) = 2 \sin t \), and for z, it is constant at \( z(t) = 1 \). Because these functions trace the path of a point as t varies, they can describe more complex shapes than Cartesian equations, like the loop or spiral our curve might represent.

Parametric equations are especially useful when dealing with intersections, as they allow us to hone in on specific points on a curve by finding the corresponding t values.

The sine function is a fundamental trigonometric function that oscillates between -1 and 1. It's periodic, with a standard period of \(2\pi\) radians, and its graph forms a wave-like pattern. This function is often written as \(\sin(t)\), where t is typically an angle measured in radians.

Understanding the behavior of the sine function is crucial for solving our intersection problem. At the heart of this sine wave are special angles where the sine values are well-known; for instance, \(\sin(\pi/6) = 1/2\), \(\sin(\pi/2) = 1\), and \(\sin(\pi) = 0\). By comparing the wave's pattern to the unit circle, we can determine that the sine function has positive values in the first and second quadrants, which correspond to angles between 0 and \(\pi\).

This characteristic of the sine function is essential for determining where the plane y = 1 intersects with our parametric curve, as it highlights the specific conditions required for the y-component to reach the height of 1.

Arcsin, or inverse sine, is the function that tells us which angle t has a given sine value. In our example, we're particularly interested in the angle whose sine is \(1/2\). Mathematically, we express this as: \[t = \arcsin\left(\frac{1}{2}\right)\]The standard output for arcsin is an angle in the range \([-\pi/2, \pi/2]\), or the first and fourth quadrants of the unit circle. However, the sine function is positive in both the first and second quadrants. To find the angles where \(\sin(t) = 1/2\) and not just the primary arc, we need to also consider \(t = \pi - \arcsin(1/2)\), which lies in the second quadrant.

It's essential to remember that when we solve for an angle using arcsin, especially in the context of geometry or trigonometry problems involving parametric equations or curve intersections, we may need to find more than one angle to fully address the problem. That's why, in the exercise, the determination of the two t values (\(t_1\) and \(t_2\)) is a decisive step for finding all points of intersection.