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Converting parametric equations into a rectangular form is an important skill in mathematics. Parametric equations express the coordinates of points on a curve as functions of a parameter, typically denoted by \( \theta \) or \( t \). For example, the given parametric equations are \( x = 4\cos\theta \) and \( y = 3\sin\theta \). The goal is to express these relationships purely in terms of \( x \) and \( y \).

To achieve this, you set up equations that relate \( x \) and \( y \) without the parameter. You use known identities or relationships, such as trigonometric identities, to transform the parametric equations. Once the equations involve only \( x \) and \( y \), the curve is represented in rectangular form.

• Identify the type of conic section, like circles, ellipses, or hyperbolas.
• Use algebraic manipulation to remove the parameter.
• Express the equation in standard rectangular form.

An ellipse is a set of points such that the sum of the distances to two fixed points (called foci) is constant. The standard equation of an ellipse centered at the origin is \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] where \( a \) and \( b \) are the semi-major and semi-minor axes. The given problem resulted in the equation \[ \frac{x^2}{16} + \frac{y^2}{9} = 1 \].

Here, the values \( a^2 = 16 \) and \( b^2 = 9 \) correspond to \( a = 4 \) and \( b = 3 \). This tells us:

• The semi-major axis is 4 units, lying along the x-axis since \( a > b \).
• The semi-minor axis is 3 units, lying along the y-axis.
• When \( a = b \), the ellipse becomes a circle.

Understanding these properties helps in sketching and analyzing the ellipse's graph.

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variables. A crucial identity is the Pythagorean identity: \( \cos^2\theta + \sin^2\theta = 1 \). This identity helps eliminate a parameter, like \( \theta \), from equations.

In the provided exercise, you solve for \( \cos\theta \) and \( \sin\theta \) using the parametric equations:

• \( \cos\theta = \frac{x}{4} \)
• \( \sin\theta = \frac{y}{3} \)

Substituting these into the Pythagorean identity gives \[ \left(\frac{x}{4}\right)^2 + \left(\frac{y}{3}\right)^2 = 1 \].

Using identities effectively simplifies complex parametric forms into simpler rectangular forms.

The domain of a function denotes all the possible input values (or x-values) for which the function is defined and returns a real number. When converting parametric equations into a rectangular form, identifying the domain is essential.

Based on the original parametric equations \( x = 4\cos\theta \) and \( y = 3\sin\theta \), the \( \theta \) varies between \( 0 \) and \( 2\pi \), cyclically covering all values of \( \cos\theta \) from -1 to 1 and \( \sin\theta \) from -1 to 1.

• This results in \( x \) ranging from \(-4\) to \(4\).
• Similarly, \( y \) ranges from \(-3\) to \(3\).

The domain of the equation \( \frac{x^2}{16} + \frac{y^2}{9} = 1 \) accommodates these ranges. Thus, it's all values of \( x \) between \(-4 \leq x \leq 4\), consistent with the range of a parameterized ellipse.