91Ó°ÊÓ

In order to transform parametric equations into a rectangular equation, knowing some fundamental trigonometric identities is crucial. One key identity we often use is the Pythagorean identity, \( \cos^2(t) + \sin^2(t) = 1 \). This allows us to relate cosine and sine functions to each other. For example, if we isolate \( \cos(t) \) and \( \sin(t) \) from the given parametric equations, we can substitute them into the Pythagorean identity.
This process lets us eliminate the parameter \( t \,\). By doing so, we convert the parametric equations into a single rectangular equation. This crucial step simplifies our task of understanding and graphing the associated curve.

A rectangular equation describes a relationship between \( x \) and \( y \,\) without involving a parameter \( t \.\) To find a rectangular equation from parametric equations, we first isolate the trigonometric functions, as shown:
From \( x = 1 + 2 \cos(t)\,\):
\(\cos(t) = \frac{x-1}{2}\,\)
From \( y = 2 + 2 \sin(t)\,\):
\(\sin(t) = \frac{y-2}{2}\,\)
We then substitute these expressions for \( \cos(t) \) and \( \sin(t) \) into the Pythagorean identity:
\[ \left( \frac{x-1}{2} \right)^2 + \left( \frac{y-2}{2} \right)^2 = 1 \]
Simplifying gives:
\[ (x-1)^2 + (y-2)^2 = 4 \,\]
This is the rectangular form of the given parametric equations.

Graphing parametric equations involves plotting points that correspond to different values of the parameter \( t \,.\) As \( t \) varies from \( 0 \) to \( 2\pi \,,\) each pair \( (x,y) \) represents a point on the curve.
For the given parametric equations \[ x = 1 + 2 \cos(t)\,\] and \[ y = 2 + 2 \sin(t)\,\], start by choosing values of \( t \). Calculate corresponding \( x \) and \( y \) values, and plot these points on a coordinate plane.
Once you have enough points, connect them smoothly to reveal the shape of the curve. In this case, the points form a circle centered at \( (1, 2) \) with a radius of 2. This helps visualize how the curve evolves as the parameter \( t \) changes.

A circle's equation in the rectangular form is \[ (x-h)^2 + (y-k)^2 = r^2 \,\] where \( (h,k) \) is the center and \( r \) is the radius.
From our rectangular equation \[ (x-1)^2 + (y-2)^2 = 4 \,\] we identify:

• Center: \( (1, 2) \.\)
• Radius: \( \sqrt{4} = 2 \,.\)

This gives us a clear representation of the circle described by the parametric equations.
Knowing this form is essential because it simplifies graphing the curve and understanding its geometric properties.
By converting parametric equations into this format, we gain insight into the nature of the curve, making problems easier to solve and visualize.