Question

The pairs of parametric equations represent lines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents. $$ \begin{aligned} &x=3 \cos t \\ &y=4 \sin t \end{aligned} $$

Short answer

The curve is an ellipse.

Step-by-step solution

Identify the Parametric Equations

We are given the parametric equations \( x = 3 \cos t \) and \( y = 4 \sin t \). These types of equations are commonly used to express curves in a plane, utilizing a parameter \( t \). Our task initially is to identify what kind of curve they describe.

Recognize the Structure of an Ellipse

The equations \( x = a \cos t \) and \( y = b \sin t \) are the general form of the parametric equations for an ellipse. Here, \( a \) and \( b \) are the semi-major and semi-minor axes, respectively. In our given equations, \( a = 3 \) and \( b = 4 \), which indicate that the curve is an ellipse with semi-major axis 4 and semi-minor axis 3.

Confirm the Curve Type

To confirm this, we can eliminate the parameter \( t \) by expressing both equations as a function of \( \theta \) and use the Pythagorean identity \( \sin^2 t + \cos^2 t = 1 \). Rewriting \( x = 3 \cos t \) gives \( \cos t = \frac{x}{3} \) and \( y = 4 \sin t \) gives \( \sin t = \frac{y}{4} \). Substitute into \( \sin^2 t + \cos^2 t = 1 \), leading to \( \left(\frac{x}{3}\right)^2 + \left(\frac{y}{4}\right)^2 = 1 \). This is the standard equation of an ellipse centered at the origin.

Key concepts

Ellipse

An ellipse is a type of curve on a plane surrounding two focal points. For any point on the ellipse, the sum of the distances to the two foci is constant. Ellipses are popular in geometry due to their unique properties and applications in real life, such as planetary orbits.

The general equation for an ellipse in parametrized form is:

• \( x = a \cos t \)
• \( y = b \sin t \)

Here, \( a \) is the length of the semi-major axis (half the longest diameter), and \( b \) is the length of the semi-minor axis (half the shortest diameter).

In our exercise, the parametric equations were given as \( x = 3 \cos t \) and \( y = 4 \sin t \). From this, we conclude that the values \( a = 3 \) and \( b = 4 \) are the lengths of the semi-minor and semi-major axes, respectively. This tells us the ellipse is oriented along the "y" axis with the center at the origin.

It's important to recognize that such equations reveal more than just the type of curve. They show orientations and dimensions, crucial for graphing or interpreting data within various fields.

Parametric Curves

Parametric curves allow us to describe both simple and complex curves using a parameter, commonly represented as \( t \). This method of presentation offers flexibility, enabling an easy depiction of curves that might not be as straightforward when described by traditional Cartesian coordinates.

Imagine having a pen tied to a rotating wheel tracing a path on paper. Adjusting the wheel's position or rotation will trace different curves—parametric equations use this principle by tying the \( x \) and \( y \) coordinates to a common parameter \( t \).

By utilizing parametric equations, like those in the given example \( x = 3 \cos t \) and \( y = 4 \sin t \), we can precisely describe an ellipse. The parameter \( t \) typically represents an acute angle in trigonometric terms, but it could even extend to time in dynamic systems, enhancing our capability to model the real world.

This makes parametric forms highly versatile, fitting scenarios ranging from mechanical systems to computer graphics, where plotting point-by-point might be needed for complex animations or simulations.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for every value of the occurring variable. They serve as essential tools for simplifying expressions and solving trigonometric equations.

One of the most fundamental identities used often is the Pythagorean identity:

• \( \sin^2 t + \cos^2 t = 1 \)

This identity is foundational in mathematics, forming the basis for analyzing various curves, including ellipses.

When solving the given parametric equations \( x = 3 \cos t \) and \( y = 4 \sin t \), recognizing which identity applies allows for simplifying and transforming equations into more recognizable Cartesian forms. Through substitution and manipulation aided by \( \sin^2 t + \cos^2 t = 1 \), the equation of an ellipse was reached:

• \( \left(\frac{x}{3}\right)^2 + \left(\frac{y}{4}\right)^2 = 1 \)

Trigonometric identities, like this, are used beyond the classroom, forming the bedrock of computations in physics, engineering, and many other fields.