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. $$ \begin{aligned} &x=2 \cos (3 t) \\ &y=5 \sin (3 t) \end{aligned} $$

Short answer

The given parametric equations represent an ellipse.

Step-by-step solution

Identify the Structure

The given parametric equations are similar to the ones used to describe circular and elliptical paths. They are:\[ x = 2 \cos(3t) \] and \[ y = 5 \sin(3t) \].

Analyze the Parametric Equations

The general parametric form of an ellipse is \( x = a \cos(t) \) and \( y = b \sin(t) \), where \( a \) and \( b \) are the semi-major and semi-minor axes, respectively. Here, the parametric equations can be rewritten as \( x = 2 \cos(3t) \) and \( y = 5 \sin(3t) \), indicating an ellipse.

Confirm the Transformation

Since the equations have the form of \( x = a \cos(At) \) and \( y = b \sin(At) \), where \( A \) is a common multiplier for \( t \), this also fits the general description of an ellipse with a rotational speed modification but still retains the elliptical form. The constants \( 2 \) and \( 5 \) serve as semi-major and semi-minor axes, confirming our curve as an ellipse.

Key concepts

Ellipses

When we deal with ellipses in parametric equations, we see a set structure that helps us recognize these elegant curves. An ellipse can be defined by its horizontal and vertical stretches, represented by the semi-major axis (ae) and the semi-minor axis (be). The standard parametric form of an ellipse is \( x = a \cos(t) \) and \( y = b \sin(t) \)​. This structure forms a stretched circular shape, known as an ellipse.

In the problem at hand, our equations \( x = 2 \cos(3t) \) and \( y = 5 \sin(3t) \) fit this form. Here:

• \( a = 2 \), which is the semi-minor axis, indicating the extent of the ellipse horizontally.
• \( b = 5 \), the semi-major axis, controlling the vertical stretch.

The elliptical path does not change if \( t \) is multiplied by a factor (in this case, 3), solely affecting the speed of traversal. The fundamental shape remains that of an ellipse.

Trigonometric Functions

Trigonometric functions such as sine and cosine play a crucial role in parametric equations representing curves. Sine (\sin) and cosine (\cos) functions fluctuate between -1 and 1, which is essential when modelling cyclical or circular patterns as seen in many natural phenomena.

The general equations use these functions to explain the variation in direction and distance at any given point along the curve. For \( x = a \cos(t) \) and \( y = b \sin(t) \):

• The function \( \cos(t) \) contributes to the x-coordinate, creating horizontal movement in conjunction with constant \( a \).
• The function \( \sin(t) \) contributes to the y-coordinate, resulting in vertical motion with constant \( b \).

For our example, the coefficients 2 and 5 scale cosine and sine, stretching the ellipse along the respective axes. Also, both \( \cos(3t) \) and \( \sin(3t) \) suggest the cycle is completed three times more quickly than the standard units.

Curve Identification

Identifying geometric curves formed by parametric equations involves matching formulas with shapes such as circles, ellipses, and other conics. Recognizing these patterns is an essential skill in geometry and calculus.

In particular:

• **Lines**: Often represented in linear parametric form \( x = a + bt \), \( y = c + dt \).
• **Parabolas**: Take different forms based on axis symmetry and involve squares of one parameter like \( x = at^2 \), \( y = bt + c \).
• **Ellipses and Circles**: Ellipses match the pattern \( x = a \cos(t) \), \( y = b \sin(t) \), with circles being a special case where \( a = b \).
• **Hyperbolas**: Typically shown in \( x = a \sec(t) \), \( y = b \tan(t) \).

In our situation, recognizing that the structure of \( x = 2 \cos(3t) \) and \( y = 5 \sin(3t) \) adheres to the ellipse form is key to identifying the curve as an ellipse rather than another conic section. These recognizable properties allow us to categorize the equations appropriately.