📐 Level 3 - Topic 6: Parametric Equations & Trigonometry - Curves in Motion 🚀✏️

1) Introduction to Parametric Equations - Describing Motion and Curves 🚀

Welcome to Level 3 - Topic 6: Parametric Equations and Trigonometry. So far, we've described curves using equations in terms of \( x \) and \( y \) (Cartesian) or \( r \) and \( \theta \) (Polar). Now, we introduce a new way to describe curves using parametric equations. This method is particularly useful for describing motion and complex curves, and it has a deep connection with trigonometry.

What are Parametric Equations?

Parameter Introduction: Instead of directly relating \( y \) to \( x \), or \( r \) to \( \theta \), parametric equations define both \( x \) and \( y \) as functions of a third variable, say \( t \), called a parameter. Typically, \( t \) represents time, angle, or some other independent variable.
Equations Form: A set of parametric equations is usually given as: \[ \begin{cases} x = f(t) \\ y = g(t) \end{cases} \] For each value of \( t \), we get a point \( (x, y) \) on the curve. As \( t \) varies, the point \( (x, y) = (f(t), g(t)) \) traces out a curve in the Cartesian plane.
Describing Motion: Parametric equations are excellent for describing the path of a moving object. If \( t \) is time, then \( (x(t), y(t)) \) gives the position of the object at time \( t \).

Why Trigonometry in Parametric Equations?

Periodic Motion: Trigonometric functions (sine, cosine) are naturally periodic, making them ideal for describing periodic motion like circular or oscillatory movements.
Circular and Elliptical Paths: Curves like circles, ellipses, cycloids, and epicycloids are easily and elegantly represented using trigonometric parametric equations.
Simplifying Complex Curves: Many complex curves become simpler to express and analyze in parametric form using trigonometric functions.

In this topic, we will explore how to use trigonometric functions to define parametric equations, understand the curves they represent, and analyze their properties. Let's begin our journey into the world of curves in motion! 🚀📐✏️

2) Basic Parametric Curves: Lines, Circles, and Ellipses ➖⏺️<0xE2><0x97><0x8A>

Let's start by representing basic geometric shapes using parametric equations, particularly focusing on how trigonometric functions play a role.

Parametric Equations for Basic Curves

Line Segment: To parameterize a line segment from \( (x_1, y_1) \) to \( (x_2, y_2) \), we can use: \[ \begin{cases} x = x_1 + (x_2 - x_1)t \\ y = y_1 + (y_2 - y_1)t \end{cases}, \quad 0 \leq t \leq 1 \] Here, \( t=0 \) gives \( (x_1, y_1) \) and \( t=1 \) gives \( (x_2, y_2) \). For a line, we can let \( t \) range over all real numbers.
Circle: A circle centered at \( (h, k) \) with radius \( r \) can be parameterized using trigonometric functions: \[ \begin{cases} x = h + r \cos t \\ y = k + r \sin t \end{cases}, \quad 0 \leq t \leq 2\pi \] Here, \( t \) is the angle measured counterclockwise from the positive x-direction. As \( t \) varies from 0 to \( 2\pi \), we trace the circle once.
Ellipse: An ellipse centered at \( (h, k) \) with semi-axes \( a \) and \( b \) (horizontal and vertical respectively) can be parameterized as: \[ \begin{cases} x = h + a \cos t \\ y = k + b \sin t \end{cases}, \quad 0 \leq t \leq 2\pi \] If \( a = b = r \), it reduces to a circle.

Example 1: Parametric Equations of a Circle

Find parametric equations for a circle centered at \( (2, -1) \) with radius 3.

1. Identify parameters: Center \( (h, k) = (2, -1) \), radius \( r = 3 \).
2. Apply circle parametric form: Use the standard parametric equations for a circle: \[ \begin{cases} x = h + r \cos t \\ y = k + r \sin t \end{cases} \]
3. Substitute values: Substitute \( h = 2, k = -1, r = 3 \): \[ \begin{cases} x = 2 + 3 \cos t \\ y = -1 + 3 \sin t \end{cases}, \quad 0 \leq t \leq 2\pi \]

Solution: Parametric equations are \( x = 2 + 3 \cos t \), \( y = -1 + 3 \sin t \), for \( 0 \leq t \leq 2\pi \).

Example 2: Parametric Equations of an Ellipse

Find parametric equations for an ellipse centered at the origin, with semi-major axis 5 along the x-axis and semi-minor axis 2 along the y-axis.

1. Identify parameters: Center \( (h, k) = (0, 0) \), semi-major axis \( a = 5 \), semi-minor axis \( b = 2 \).
2. Apply ellipse parametric form: Use the standard parametric equations for an ellipse: \[ \begin{cases} x = h + a \cos t \\ y = k + b \sin t \end{cases} \]
3. Substitute values: Substitute \( h = 0, k = 0, a = 5, b = 2 \): \[ \begin{cases} x = 5 \cos t \\ y = 2 \sin t \end{cases}, \quad 0 \leq t \leq 2\pi \]

Solution: Parametric equations are \( x = 5 \cos t \), \( y = 2 \sin t \), for \( 0 \leq t \leq 2\pi \).

Example 3: Parametric Equations of a Line Segment

Find parametric equations for the line segment from \( (1, 2) \) to \( (4, 6) \).

1. Identify points: \( (x_1, y_1) = (1, 2) \), \( (x_2, y_2) = (4, 6) \).
2. Apply line segment parametric form: Use the standard parametric equations for a line segment: \[ \begin{cases} x = x_1 + (x_2 - x_1)t \\ y = y_1 + (y_2 - y_1)t \end{cases}, \quad 0 \leq t \leq 1 \]
3. Substitute values: Substitute \( x_1 = 1, y_1 = 2, x_2 = 4, y_2 = 6 \): \[ \begin{cases} x = 1 + (4 - 1)t = 1 + 3t \\ y = 2 + (6 - 2)t = 2 + 4t \end{cases}, \quad 0 \leq t \leq 1 \]

Solution: Parametric equations are \( x = 1 + 3t \), \( y = 2 + 4t \), for \( 0 \leq t \leq 1 \).

3) Calculus with Parametric Equations - Derivatives and Tangents 📈✏️

Calculus can be applied to parametric equations to find slopes of tangent lines, concavity, and more. Here we focus on finding derivatives to determine tangent lines to parametrically defined curves.

Derivatives for Parametric Equations

If a curve is given by parametric equations \( x = f(t) \) and \( y = g(t) \), and if \( \frac{dx}{dt} \neq 0 \), then the derivative \( \frac{dy}{dx} \) is given by:

\( \frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{g'(t)}{f'(t)} \)

This formula gives the slope of the tangent line to the parametric curve at the point \( (f(t), g(t)) \), provided \( f'(t) \neq 0 \).

Example 4: Slope of Tangent Line to a Parametric Circle

Find the slope of the tangent line to the circle \( x = 3 \cos t \), \( y = 3 \sin t \) at \( t = \frac{\pi}{4} \).

1. Find \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \): \[ \frac{dx}{dt} = \frac{d}{dt}(3 \cos t) = -3 \sin t, \quad \frac{dy}{dt} = \frac{d}{dt}(3 \sin t) = 3 \cos t \]
2. Calculate \( \frac{dy}{dx} \): \[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{3 \cos t}{-3 \sin t} = -\cot t \]
3. Evaluate \( \frac{dy}{dx} \) at \( t = \frac{\pi}{4} \): \[ \left. \frac{dy}{dx} \right|_{t=\pi/4} = -\cot \left( \frac{\pi}{4} \right) = -1 \]

Solution: The slope of the tangent line at \( t = \frac{\pi}{4} \) is -1.

Example 5: Equation of Tangent Line to a Parametric Curve

Find the equation of the tangent line to the curve \( x = t^2 \), \( y = t^3 - 3t \) at \( t = 2 \).

1. Find point \( (x, y) \) at \( t = 2 \): \[ x = (2)^2 = 4, \quad y = (2)^3 - 3(2) = 8 - 6 = 2. \quad \text{Point is } (4, 2). \]
2. Find \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \): \[ \frac{dx}{dt} = 2t, \quad \frac{dy}{dt} = 3t^2 - 3 \]
3. Calculate \( \frac{dy}{dx} \): \[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{3t^2 - 3}{2t} \]
4. Evaluate \( \frac{dy}{dx} \) at \( t = 2 \): \[ \left. \frac{dy}{dx} \right|_{t=2} = \frac{3(2)^2 - 3}{2(2)} = \frac{12 - 3}{4} = \frac{9}{4}. \quad \text{Slope } m = \frac{9}{4}. \]
5. Equation of tangent line using point-slope form \( y - y_1 = m(x - x_1) \): \[ y - 2 = \frac{9}{4}(x - 4) \Rightarrow y = \frac{9}{4}x - 9 + 2 \Rightarrow y = \frac{9}{4}x - 7 \]

Solution: The equation of the tangent line is \( y = \frac{9}{4}x - 7 \).

4) Cycloids and Related Curves - Parametric Trigonometry in Action 🌀🔄

Trigonometric parametric equations are particularly effective in describing curves generated by rolling circles. Let's explore cycloids and related curves.

Cycloids and Related Curves

Cycloid: The path traced by a point on the circumference of a circle as it rolls along a straight line. If a circle of radius \( r \) rolls along the x-axis, and starts with a point at the origin, the parametric equations of the cycloid are: \[ \begin{cases} x = r(t - \sin t) \\ y = r(1 - \cos t) \end{cases}, \quad t \in \mathbb{R} \] Here, \( t \) is the angle of rotation of the circle in radians.
Trochoid: A generalization of the cycloid. It is the path traced by a point at a fixed distance \( d \) from the center of a circle of radius \( r \) as the circle rolls along a straight line. If \( d < r \), we get a curtate trochoid (loopless); if \( d > r \), a prolate trochoid (with loops); if \( d = r \), a cycloid. Parametric equations: \[ \begin{cases} x = rt - d \sin t \\ y = r - d \cos t \end{cases}, \quad t \in \mathbb{R} \]
Epicycloid: The path traced by a point on the circumference of a circle rolling around another fixed circle.
Hypocycloid: The path traced by a point on the circumference of a circle rolling inside another fixed circle.

Example 6: Parametric Equations of a Cycloid with Radius 2

Write parametric equations for the cycloid generated by a circle of radius 2 rolling along the x-axis.

1. Identify radius: Radius \( r = 2 \).
2. Apply cycloid parametric form: Use the standard parametric equations for a cycloid: \[ \begin{cases} x = r(t - \sin t) \\ y = r(1 - \cos t) \end{cases} \]
3. Substitute radius value: Substitute \( r = 2 \): \[ \begin{cases} x = 2(t - \sin t) \\ y = 2(1 - \cos t) \end{cases}, \quad t \in \mathbb{R} \]

Solution: Parametric equations are \( x = 2(t - \sin t) \), \( y = 2(1 - \cos t) \), for \( t \in \mathbb{R} \).

Example 7: Finding a Point on a Cycloid

For the cycloid \( x = 3(t - \sin t) \), \( y = 3(1 - \cos t) \), find the coordinates of the point when the circle has rotated by \( \theta = \frac{\pi}{2} \) radians.

1. Recognize parameter meaning: In cycloid equations, \( t \) represents the angle of rotation, so \( t = \theta = \frac{\pi}{2} \).
2. Substitute \( t = \frac{\pi}{2} \) into equations: \[ x = 3\left(\frac{\pi}{2} - \sin \frac{\pi}{2}\right) = 3\left(\frac{\pi}{2} - 1\right) = \frac{3\pi}{2} - 3 \] \[ y = 3\left(1 - \cos \frac{\pi}{2}\right) = 3(1 - 0) = 3 \]

Solution: The coordinates are \( \left( \frac{3\pi}{2} - 3, 3 \right) \approx (1.71, 3) \).

5) Converting Between Parametric, Cartesian, and Polar Equations 🔄

It's useful to be able to convert between parametric, Cartesian, and polar forms of equations to understand curves from different perspectives.

Conversions Between Equation Forms

Parametric to Cartesian: Eliminate the parameter \( t \) from the parametric equations \( x = f(t) \) and \( y = g(t) \) to get a Cartesian equation in \( x \) and \( y \). This is not always straightforward and possible algebraically, but often can be done if you can solve for \( t \) from one equation and substitute it into the other.
Cartesian to Parametric: To parameterize a Cartesian equation \( y = F(x) \), you can often simply set \( x = t \), then \( y = F(t) \). For more complex forms, or to use trigonometric parameters, transformations or geometric understanding is needed.
Parametric to Polar: Use the relationships \( x = r \cos \theta \) and \( y = r \sin \theta \). Substitute \( x = f(t) \) and \( y = g(t) \) to get \( r \cos \theta = f(t) \) and \( r \sin \theta = g(t) \). You might need to solve for \( r \) and \( \theta \) in terms of \( t \), or express relationships in polar form.
Polar to Parametric: Given a polar equation \( r = h(\theta) \), use the conversion formulas \( x = r \cos \theta \) and \( y = r \sin \theta \). Replace \( r \) with \( h(\theta) \). Treat \( \theta \) as the parameter \( t \), then \( x = h(t) \cos t \) and \( y = h(t) \sin t \) are parametric equations.

Example 8: Parametric to Cartesian Conversion (Circle)

Convert the parametric equations \( x = \cos t \), \( y = \sin t \) to a Cartesian equation.

1. Use trigonometric identity: We know \( \cos^2 t + \sin^2 t = 1 \).
2. Square and add parametric equations: Square both equations: \( x^2 = \cos^2 t \), \( y^2 = \sin^2 t \).
3. Combine: Add the squared equations: \( x^2 + y^2 = \cos^2 t + \sin^2 t = 1 \).

Solution: The Cartesian equation is \( x^2 + y^2 = 1 \), a unit circle centered at the origin.

Example 9: Cartesian to Parametric Conversion (Parabola)

Find parametric equations for the parabola \( y = x^2 \).

1. Choose parameter: Let \( x = t \) be the parameter.
2. Substitute into Cartesian equation: Substitute \( x = t \) into \( y = x^2 \) to get \( y = t^2 \).

Solution: Parametric equations are \( x = t \), \( y = t^2 \).

Example 10: Polar to Parametric Conversion (Spiral)

Convert the polar equation \( r = \theta \) to parametric equations.

1. Use conversion formulas: \( x = r \cos \theta \), \( y = r \sin \theta \).
2. Substitute \( r = \theta \): Replace \( r \) with \( \theta \) in the formulas: \( x = \theta \cos \theta \), \( y = \theta \sin \theta \).
3. Use parameter \( t = \theta \): Let \( \theta = t \).

Solution: Parametric equations are \( x = t \cos t \), \( y = t \sin t \).

6) Practice Questions 🎯

6.1 Fundamental – Parametric Equations Basics

1. Find parametric equations for a circle centered at \( (-1, 2) \) with radius 4.

2. Find parametric equations for an ellipse centered at \( (0, 0) \) with semi-axes 3 and 5 (major along y-axis).

3. Find parametric equations for the line segment from \( (-2, 3) \) to \( (1, -1) \).

4. For the parametric curve \( x = 2t - 1 \), \( y = t^2 + 1 \), find \( \frac{dy}{dx} \).

5. Find the slope of the tangent line to the circle \( x = 5\cos t \), \( y = 5\sin t \) at \( t = \frac{\pi}{3} \).

6. Write parametric equations for a cycloid generated by a circle of radius 3.

7. Convert the parametric equations \( x = 2\cos t \), \( y = 3\sin t \) to a Cartesian equation.

8. Find parametric equations for the Cartesian equation \( y = x^3 \).

9. Convert the polar equation \( r = 2\cos \theta \) to parametric equations.

10. For the cycloid \( x = r(t - \sin t) \), \( y = r(1 - \cos t) \), what are the coordinates when \( t = \pi \)?

6.2 Challenging – Advanced Parametric Concepts and Conversions 💪🚀

1. Find the equation of the tangent line to the curve \( x = \sin(2t) \), \( y = \cos(t) \) at \( t = \frac{\pi}{2} \).

2. Derive the Cartesian equation of the curve given by \( x = \sec t \), \( y = \tan t \). Identify the curve.

3. Parameterize the upper half of the circle \( x^2 + y^2 = 9 \) starting at \( (3, 0) \) and going counterclockwise.

4. Consider a trochoid with \( r = 2 \) and \( d = 3 \). Write its parametric equations and describe its loops.

5. (Conceptual) Explain why parametric equations are more versatile than Cartesian equations for describing motion. Give an example where parametric form is significantly simpler to use than Cartesian.

7) Summary - Parametric Equations & Trigonometry - Describing Curves in Motion 🎉

Parametric Equations: Define \( x \) and \( y \) as functions of a parameter \( t \), i.e., \( x = f(t) \), \( y = g(t) \), to describe curves.
Trigonometry in Parametric Equations: Trigonometric functions are used to parameterize circles, ellipses, cycloids, and other periodic curves.
Basic Parametric Curves: Lines, circles, ellipses have standard trigonometric parametric forms.
Calculus with Parametric Equations: Derivative \( \frac{dy}{dx} = \frac{dy/dt}{dx/dt} \) allows finding tangent lines.
Cycloids and Trochoids: Examples of curves naturally described by trigonometric parametric equations, representing rolling circle paths.
Conversions: Ability to convert between parametric, Cartesian, and polar forms enhances understanding of curves from different perspectives.
Versatility: Parametric equations are powerful for describing motion, complex shapes, and curves where Cartesian form is cumbersome.

Congratulations! You have now explored parametric equations and their deep connection with trigonometry. You learned to represent basic shapes parametrically, apply calculus to parametric curves, understand cycloids, and convert between equation forms. This topic expands your toolkit for describing and analyzing curves, especially those related to motion and periodic phenomena. Next, we will move to Level 3 - Topic 7! 🚀📐✏️🌀🌟