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The Pythagorean Identity is a fundamental concept in trigonometry. It is given by the equation \( \sin^2 t + \cos^2 t = 1 \). This identity asserts that for any angle \( t \), the sum of the square of the sine of \( t \) and the square of the cosine of \( t \) is always equal to 1.

This relationship is incredibly useful, especially when dealing with parametric equations. By expressing both \( \sin t \) and \( \cos t \) in terms of other variables, you can eliminate the parameter \( t \) completely from the equations. For example, if you have \( \sin t = \frac{x - 3}{2} \) and \( \cos t = \frac{y - 1}{2} \), you can substitute these into the Pythagorean Identity:

• \[\left(\frac{x - 3}{2}\right)^2 + \left(\frac{y - 1}{2}\right)^2 = 1\]

By using this identity, you can relate \( x \) and \( y \) directly, removing the parameter \( t \) completely.

Understanding how to graph circles is an essential part of geometry and algebra. When you have an equation of the form \((x - h)^2 + (y - k)^2 = r^2\), it represents a circle with:

• Center at \((h, k)\)
• Radius \( r \)

This is precisely what happened when we simplified the parametric equations to the form \((x - 3)^2 + (y - 1)^2 = 4\). It shows that the circle is centered at \((3,1)\) with a radius of 2, because \(r = \sqrt{4} = 2\).

To graph this circle, you would start by marking the center point on the graph. Then, measure a distance of 2 units in all directions from the center to draw the circle. This involves:

• Going 2 units up, down, left, and right from the center.
• Sketching a round curve that connects these boundaries.

Graphing circles can visualize solutions to problems involving radial distances and centers, making geometric understanding clearer.

Eliminating the parameter from parametric equations is a common task in algebra, which helps convert a problem from parametric to standard form. This process simplifies analysis and graphing.

For instance, consider the parametric equations \(x = 3 + 2 \sin t\) and \(y = 1 + 2 \cos t\). To eliminate \(t\), follow these steps:

• First, solve each equation for \(\sin t\) and \(\cos t\):
  • \(\sin t = \frac{x - 3}{2}\)
  • \(\cos t = \frac{y - 1}{2}\)


• Next, apply the Pythagorean Identity \(\sin^2 t + \cos^2 t = 1\) by substituting the expressions for \(\sin t\) and \(\cos t\):
  • \[\left(\frac{x - 3}{2}\right)^2 + \left(\frac{y - 1}{2}\right)^2 = 1\]


• Finally, simplify to obtain \((x - 3)^2 + (y - 1)^2 = 4\).

This process not only removes \(t\) but reveals the intrinsic relationship between \(x\) and \(y\), allowing for a more straightforward graphing and analysis of the problem.