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

Short answer

The parametric equations represent a circle with radius 3.

Step-by-step solution

Identify the Parametric Equations

The given parametric equations are \( x = 3 \cos t \) and \( y = 3 \sin t \). These equations express \(x\) and \(y\) in terms of the parameter \(t\).

Recognize Trigonometric Form

The form of the equations \(x = a \cos t \) and \(y = b \sin t \) is characteristic of a trigonometric expression. Here, both amplitudes \(a\) and \(b\) are equal to 3.

Analyze the Amplitudes

Since the coefficients of \( \cos t \) and \( \sin t \) (i.e., 3 and 3 respectively) are equal, it implies that the figure is symmetric with respect to both axes.

Deduce the Type of Curve

In parametric equations, if \( x = a \cos t \) and \( y = b \sin t \) with equal amplitudes \(a = b\), the equations represent a circle. The radius of this circle is the common coefficient, which is 3.

Key concepts

Trigonometric Form

Trigonometric form in parametric equations is an essential concept that connects trigonometry with algebraic expressions of geometric figures. When you see equations like \( x = a \cos t \) and \( y = b \sin t \), these are expressed in trigonometric form. Here, \( a \) and \( b \) function as the amplitudes of the trigonometric functions. This form is often used to describe curves such as circles and ellipses.

Understanding the trigonometric form means recognizing these equations are derived from the basic identities of sine and cosine, which are periodic functions. Their periodic nature helps in generating patterns like circles when combined with parameters like \( t \), which represents the angle in radians. This angle, \( t \), varies typically from 0 to \( 2\pi \) to complete a full revolution, forming standard circular paths.

Symmetry in Parametric Curves

Symmetry can reveal a lot about a parametric curve's geometric properties. When examining parametric equations like \( x = a \cos t \) and \( y = b \sin t \), if \( a \) and \( b \) are equal, the curve enjoys perfect symmetry. This is because each coordinate is scaled by the same factor, meaning every point on the curve is mirrored across the both axes.

Symmetry is more than just aesthetics; it is crucial in calculating geometry-related measures and understanding the behavior of the curve visually. The presence of symmetry in these trigonometric parametric equations ensures that as \( t \) varies, the points \( (x, y) \) cover the curve evenly without distortion, illustrating completeness and balance in shape.

Geometric Interpretation of Parametric Curves

The geometric interpretation of parametric curves translates mathematical equations into visual patterns. In the case of the parametric equations \( x = 3 \cos t \) and \( y = 3 \sin t \), the geometric shape is a circle. Parametric equations don't directly give you the entire figure; instead, they represent the motion over time that forms the curve.

As \( t \) increases from 0 to \( 2\pi \), imagine a point moving along the curve. The trigonometric functions navigate the point through the circular path, establishing a pattern dictated by the equations. This process shows that parametric equations are a tool for dynamically interpreting and drawing geometric figures, instead of static depiction like the Cartesian form.

Circle Radius Determination

Determining the radius of a circle from parametric equations is straightforward with the trigonometric form. For equations \( x = a \cos t \) and \( y = a \sin t \), the shared coefficient \( a \) is the radius. In our specific example, both equations have a coefficient of 3, confirming the radius of the circle as 3.

This relationship is derived from the Pythagorean identity \( \cos^2 t + \sin^2 t = 1 \), which maintains that the circle outlined by parametric equations always stems from this formulaic identity. By understanding that \( a = b \) and equals the radius in this scenario, you've recognized how trigonometric properties interlace with geometry to define a circle's size quantitatively.