Problem 16 - Algebra and Trigonometry

Question

\begin{array}{l}{1-22 \text { a pair of parametric equations is given. }} \\\ {\text { (a) Sketch the curve represented by the parametric equations. }} \\\ {\text { (b) Find a rectangular-coordinate equation for the curve by }} \\\ {\text { eliminating the parameter. }}\end{array} $$ x=\cos 2 t, \quad y=\sin 2 t $$

Step-by-Step Solution

Answer

The curve traces a unit circle twice with equation \(x^2 + y^2 = 1\).

Step 1: Understanding the Parametric Equations

The given parametric equations are \(x = \cos 2t\) and \(y = \sin 2t\). These are equations in terms of the parameter \(t\), where \(x\) and \(y\) represent points on a curve as \(t\) varies.

Step 2: Sketching the Parametric Curve

To sketch the curve, note that as \(t\) ranges from 0 to \(2\pi\), \(2t\) ranges from 0 to \(4\pi\). The equations represent a parametric form of the unit circle because \(\cos^2 2t + \sin^2 2t = 1\). The curve will trace the unit circle twice, completing a full circle between \(0\) and \(2\pi\) and another between \(2\pi\) and \(4\pi\).

Step 3: Converting to Rectangular Coordinates

To find a rectangular equation, start from the identity \(\cos^2 2t + \sin^2 2t = 1\). Substitute \(x = \cos 2t\) and \(y = \sin 2t\) into this identity to get \(x^2 + y^2 = 1\).

Step 4: Conclusion

The rectangular coordinate equation of the curve is \(x^2 + y^2 = 1\), representing the unit circle.

Key Concepts

Curve Sketching

Curve sketching is a technique used in mathematics to create a visual representation of a curve defined by equations. It involves plotting points of the curve by calculating the coordinates for various values of the parameter. In our case, the curve is defined by the parametric equations \(x = \cos 2t\) and \(y = \sin 2t\). To begin sketching the curve, you adjust the parameter \(t\) and calculate the corresponding \(x\) and \(y\) values.
As you vary \(t\) from 0 to \(2\pi\), the parameter \(2t\) ranges from 0 to \(4\pi\). This means that the curve will complete a full circle twice.
By using these values, students can plot the points \((\cos 2t, \sin 2t)\) on a graph. This results in a representation of a unit circle, traced twice from the start to the end of the parameter range.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are used to specify the location of a point on a plane using two perpendicular axes. These coordinates are expressed as \((x, y)\) pairs, where \(x\) represents the horizontal position, and \(y\) represents the vertical position.
In the given exercise, we started with parametric equations and converted them into a rectangular coordinate format. This process is called elimination of the parameter.
By using the trigonometric identity \(\cos^2 2t + \sin^2 2t = 1\), we substitute \(x = \cos 2t\) and \(y = \sin 2t\). Upon simplification, we derive the equation \(x^2 + y^2 = 1\), which is the rectangular equation of the unit circle. This transformation allows the curve to be studied within the familiar rectangular coordinate system.

Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin of the coordinate plane. It's a fundamental concept in trigonometry. A unit circle provides a simple way to understand how the trigonometric functions sine and cosine relate to angles.
In our problem, the parametric equations \(x = \cos 2t\) and \(y = \sin 2t\) naturally lead to the unit circle because these expressions represent the \(x\) and \(y\) coordinates of points on the circle as the angle \(2t\) varies.
The equation \(x^2 + y^2 = 1\) signifies that every point on this curve is exactly 1 unit away from the origin. The angle \(2t\) ensures that the whole circle is traced between 0 to \(2\pi\) and again from \(2\pi\) to \(4\pi\). Understanding this concept is crucial as it provides insight into various trigonometric identities and properties.

Trigonometric Identity

Trigonometric identities are equations that relate the trigonometric functions to one another. They are critical in simplifying expressions and solving equations that involve trigonometric functions.
A key trigonometric identity used in this exercise is \(\cos^2 2t + \sin^2 2t = 1\). This identity tells us that, regardless of the value of \(t\), the expression will always equal 1.
By substituting \(x = \cos 2t\) and \(y = \sin 2t\) into this identity, we simplify the parametric equations to find the rectangular coordinate equation \(x^2 + y^2 = 1\).
This step is important as it reveals the underlying geometry of the curve and aids in interpreting the mathematical behavior and properties of the curve in rectangular coordinates. Understanding these identities enhances problem-solving skills and helps in connecting different areas of mathematics seamlessly.