Problem 42 - Calculus Volume 3

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. $$ \begin{array}{l} x=3 \cos t \\ y=3 \sin t \end{array} $$

Step-by-Step Solution

Answer

The equations represent a circle with radius 3.

Step 1: Identify the Equations

The given parametric equations are:\[ x = 3 \cos t \]\[ y = 3 \sin t \]

Step 2: Recognize the Trigonometric Functions

Notice the presence of sine and cosine functions in both equations. These functions typically represent a circular motion when combined.

Step 3: Understand the Coefficients

Both coefficients of the trigonometric functions are 3, suggesting there is a constant radius involved.

Step 4: Relate Parametric Equations to a Circle Equation

Generally, when parametric equations have the form \( x = a \cos t \) and \( y = a \sin t \), they describe a circle with radius \( a \). Here \( a = 3 \). Thus, the equations describe a circle.

Key Concepts

Trigonometric Functions

Trigonometric functions are special mathematical functions that relate angles of right triangles to the ratios of two side lengths. These functions include sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)), among others. In the case of parametric equations for curves like circles, sine and cosine are often used because they provide a way to describe circular motion based on the angle of rotation.

• Sine (\( \sin \)): Given an angle \( t \) in a right triangle, \( \sin t = \frac{\text{opposite side}}{\text{hypotenuse}}\).
• Cosine (\( \cos \)): For the same angle \( t \), \( \cos t = \frac{\text{adjacent side}}{\text{hypotenuse}}\).

These functions are crucial in parametric equations for circles because they smoothly transition between values from -1 to 1, perfectly mapping out a circle. As the parameter \( t \) varies from 0 to \( 2\pi \), both sine and cosine functions trace one full revolution around the circle.

Circle Equation

When describing a circle using parametric equations, we commonly use the formulas: \[ x = a \cos t \]\[ y = a \sin t \]Here, \( a \) is the radius of the circle, and \( t \) represents the parameter, typically the angle of rotation measured in radians. This form is derived from the general equation of a circle in standard geometry, \[ x^2 + y^2 = r^2 \]where \( r \) is the radius. By squaring and adding the parametric formulas, we can derive this equation:- Substitute into the circle equation: \( (a \cos t)^2 + (a \sin t)^2 = a^2 \)- Simplified to meet \( x^2 + y^2 = a^2 \), satisfying the equation for a circle.This clear relationship between parametric and standard circle equations helps us see how parametric equations succinctly describe a simple geometric shape such as a circle using trigonometric functions and a radius.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves using algebra to study geometric problems and figures through a coordinate system. This approach allows us to utilize coordinates, such as \( (x, y) \), to describe the positions of points and shapes like lines, circles, and parabolas in a plane.Linking Parametric Equations and Geometry:Parametric equations like \[ x = 3 \cos t \] and \[ y = 3 \sin t \]provide a powerful way to describe geometries within the coordinate plane. In particular:

• Visualize Shapes: Parametric equations help visualize geometric shapes by expressing coordinates directly in terms of a parameter, offering a fluid description beyond static forms.
• Curves and Movement: They are used to describe curves and paths in terms of a moving point, typically by adjusting the parameter \( t \) to track the point's movement along a path such as a circle.

This approach makes it easier to analyze and understand the dynamics of motion and curves in the coordinate plane, catering to both static and dynamic geometric interpretations.