🔺Trigonometry Unit 12 – Parametric Equations

What Are Parametric Equations?

• Parametric equations express the coordinates of points on a curve in terms of an independent variable, typically denoted as $t$
• Consist of two equations, one for the $x$-coordinate and one for the $y$-coordinate, both in terms of $t$
  • Example: $x = \cos(t)$ and $y = \sin(t)$ for $0 \leq t \leq 2\pi$
• Allow for the representation of complex curves that may be difficult to express using a single equation in $x$ and $y$
• Provide a way to describe the motion of an object along a path, with $t$ representing time
• Enable the study of cyclical or periodic behavior, such as the motion of a pendulum or the orbit of a planet
• Offer a different perspective on curve sketching and analysis compared to traditional Cartesian equations
• Can be used to represent curves in higher dimensions by adding additional equations for each dimension

Key Concepts and Terminology

• Parameter: The independent variable, usually denoted as $t$, that controls the values of $x$ and $y$ in parametric equations
• Parametric curve: The graph of a set of parametric equations, representing the path traced by the point $(x(t), y(t))$ as $t$ varies
• Domain: The set of values that the parameter $t$ can take, often specified as an interval
• Range: The set of values that the coordinates $x$ and $y$ can take as $t$ varies over its domain
• Orientation: The direction in which a parametric curve is traced as the parameter $t$ increases
  • Clockwise orientation: The curve is traced in a clockwise direction as $t$ increases
  • Counterclockwise orientation: The curve is traced in a counterclockwise direction as $t$ increases
• Symmetry: Parametric curves can exhibit various types of symmetry, such as reflectional or rotational symmetry
• Periodicity: A parametric curve is periodic if it repeats itself at regular intervals of the parameter $t$

Graphing Parametric Equations

• To graph parametric equations, create a table of values for $t$, $x$, and $y$
  • Choose a suitable range for $t$ based on the domain specified or the problem context
  • Substitute values of $t$ into the equations for $x$ and $y$ to find corresponding coordinates
• Plot the points $(x, y)$ obtained from the table of values on a coordinate plane
• Connect the points in the order of increasing $t$ values to form the parametric curve
• Identify any key points, such as intercepts, symmetries, or points of intersection
• Determine the orientation of the curve by observing the direction in which it is traced as $t$ increases
• Use graphing technology, such as a graphing calculator or software, to visualize the curve more easily
• Analyze the behavior of the curve, such as its shape, boundedness, and any asymptotes or singularities

Converting Between Parametric and Cartesian Forms

• Parametric equations can often be converted to Cartesian form (a single equation in $x$ and $y$) by eliminating the parameter $t$
• To convert from parametric to Cartesian form:
  1. Solve one of the parametric equations for $t$ in terms of $x$ or $y$
  2. Substitute the expression for $t$ into the other parametric equation
  3. Simplify the resulting equation to obtain a Cartesian equation in $x$ and $y$
• Example: Given $x = 2t$ and $y = t^2$, solve for $t$ in the first equation: $t = \frac{x}{2}$
  • Substitute into the second equation: $y = (\frac{x}{2})^2$
  • Simplify: $y = \frac{x^2}{4}$, which is the Cartesian form
• Converting from Cartesian to parametric form is not always unique, as there may be multiple ways to parameterize a curve
• To convert from Cartesian to parametric form, introduce a parameter $t$ and express $x$ and $y$ in terms of $t$ such that the Cartesian equation is satisfied

Applications in Real-World Scenarios

• Parametric equations are used to model the motion of objects in physics and engineering
  • Example: The path of a projectile can be described using parametric equations for its horizontal and vertical positions over time
• In computer graphics and animation, parametric equations are used to generate smooth curves and surfaces
  • Bézier curves, used in vector graphics and font design, are defined using parametric equations
• Parametric equations are employed in the design of curves and surfaces in architecture and industrial design
  • Example: The profile of a car body or the shape of a building's roof can be modeled using parametric equations
• In robotics, parametric equations are used to plan and control the motion of robot arms and other mechanisms
• Parametric equations are applied in the study of planetary orbits and satellite trajectories in astronomy and space science
• In economics, parametric equations can be used to model the relationship between various economic variables, such as supply and demand curves

Common Parametric Curves

• Circle: $x = r\cos(t)$, $y = r\sin(t)$, where $r$ is the radius and $0 \leq t \leq 2\pi$
• Ellipse: $x = a\cos(t)$, $y = b\sin(t)$, where $a$ and $b$ are the semi-major and semi-minor axes, and $0 \leq t \leq 2\pi$
• Cycloid: The path traced by a point on the circumference of a circle as it rolls along a straight line
  • Equations: $x = r(t - \sin(t))$, $y = r(1 - \cos(t))$, where $r$ is the radius of the circle
• Astroid: A special case of the hypocycloid curve, resembling a four-pointed star
  • Equations: $x = a\cos^3(t)$, $y = a\sin^3(t)$, where $a$ is a constant and $0 \leq t \leq 2\pi$
• Lemniscate of Bernoulli: A figure-eight shaped curve
  • Equations: $x = \frac{a\cos(t)}{1 + \sin^2(t)}$, $y = \frac{a\sin(t)\cos(t)}{1 + \sin^2(t)}$, where $a$ is a constant
• Spiral: Various types of spirals can be represented using parametric equations
  • Example: Archimedean spiral: $x = at\cos(t)$, $y = at\sin(t)$, where $a$ is a constant

Calculus with Parametric Equations

• Differentiation: To find the derivative of a parametric curve, differentiate $x$ and $y$ with respect to $t$ separately
  • $\frac{dx}{dt}$ and $\frac{dy}{dt}$ give the rates of change of $x$ and $y$ with respect to $t$
  • The slope of the tangent line at a point on the curve is given by $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$
• Integration: To find the area under a parametric curve, integrate with respect to $t$
  • The area between the curve and the $x$-axis over the interval $[a, b]$ is given by $\int_a^b y(t) \frac{dx}{dt} dt$
  • The arc length of a parametric curve over the interval $[a, b]$ is given by $\int_a^b \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} dt$
• Parametric equations can be used to represent vector-valued functions, which are useful in physics and engineering applications
• Higher-order derivatives and integrals of parametric equations can provide information about the curvature, acceleration, and other properties of the curve

Practice Problems and Tips

• When solving problems involving parametric equations, identify the parameter and its domain
• Sketch the curve by plotting points or using graphing technology to visualize the problem
• Be comfortable converting between parametric and Cartesian forms, as some problems may require working with both representations
• When finding intersections between parametric curves, set their $x$ and $y$ equations equal and solve for the parameter $t$
• Pay attention to the orientation of the curve and any symmetries or periodicities it may have
• Practice manipulating and simplifying trigonometric and algebraic expressions that often appear in parametric equations
• When working with calculus concepts, such as derivatives and integrals, remember to apply the chain rule and substitution techniques as needed
• Analyze the behavior of the curve at key points, such as when the parameter takes on specific values or approaches infinity
• Apply parametric equations to model real-world situations, such as the motion of objects or the design of curves and surfaces