Problem 13 - Algebra and Trigonometry

Question

\(3-24=A\) pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular coordinate equation for the curve by eliminating the parameter. $$ x=2 \sin t, \quad y=2 \cos t, \quad 0 \leq t \leq \pi $$

Step-by-Step Solution

Answer

(a) The curve is a semicircle. (b) The rectangular equation is \( x^2 + y^2 = 4 \).

Step 1: Understand the Parametric Equations

The given parametric equations are \( x = 2 \sin t \) and \( y = 2 \cos t \), where \( 0 \leq t \leq \pi \). These equations define the positions \( x \) and \( y \) in terms of the parameter \( t \), which varies from 0 to \( \pi \).

Step 2: Sketch the Curve from Parametric Equations

To sketch the curve represented by the parametric equations, recognize that \( x = 2 \sin t \) and \( y = 2 \cos t \) describe a path of a point on a circle with radius 2. However, as \( t \) varies from 0 to \( \pi \), it traces the top half of the circle. The sketch will show a semicircle extending from \( (0, 2) \) to \( (0, -2) \).

Step 3: Eliminate the Parameter 't'

To find a rectangular equation, eliminate \( t \). First, square both \( x \) and \( y \): \( x^2 = (2 \sin t)^2 = 4 \sin^2 t \) and \( y^2 = (2 \cos t)^2 = 4 \cos^2 t \). Use the identity \( \sin^2 t + \cos^2 t = 1 \) to replace \( \sin^2 t + \cos^2 t = 1 \). Hence, \( \frac{x^2}{4} + \frac{y^2}{4} = 1 \).

Step 4: Write the Final Rectangular Coordinate Equation

The equation represents a circle in rectangular coordinates. Since \( \frac{x^2}{4} + \frac{y^2}{4} = 1 \) simplifies to \( x^2 + y^2 = 4 \), this is the rectangular equation for the given parametric equations for \( 0 \leq t \leq \pi \).

Step 5: Verify the Curve Portion

Check the curve portion using the parameter range \( 0 \leq t \leq \pi \). At \( t = 0 \), \( (x, y) \) is \( (0, 2) \) and at \( t = \pi \), \( (x, y) \) is \( (0, -2) \), confirming it creates the upper semicircle from \( 0 \leq x \leq \pi \).

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a way to describe the position of points on a plane using a pair of numbers. These numbers represent the horizontal and vertical distances from a fixed origin. In a rectangular coordinate system, each point is identified by an ordered pair

• \((x, y)\) where \(x\) is the distance along the horizontal axis (x-axis) and \(y\) is the distance along the vertical axis (y-axis).

This system is particularly useful because it allows us to describe geometric shapes and physical systems in a straightforward and standardized manner. By converting parametric equations, like the ones given with

• \(x = 2 \sin t\)
• \(y = 2 \cos t\)

into rectangular coordinates, we obtain a clearer understanding of the curve's shape as well as its position in the plane.

Circle

A circle is a fundamental geometric shape defined as the set of all points in a plane that are at a fixed distance (radius) from a central point (center). In the context of the given problem, the circle can be represented using the parametric equations

• \(x = 2 \sin t\)
• \(y = 2 \cos t\)

yielding the equation in rectangular coordinates \(x^2 + y^2 = 4\).
These parametric equations describe a circle because:

• The coefficients of \(\sin t\) and \(\cos t\) control the size of the circle, with the radius being 2 in this case.
• The range of the parameter \(t\), from 0 to \(\pi\), defines a semicircular path rather than a full circle, representing only the upper portion of the circle from \((0, 2)\) to \((0, -2)\).

Understanding the properties of the circle, including its radius and how it is represented in different coordinate systems, is key in analyzing and interpreting problems involving circular motion or cyclic phenomena.

Trigonometric Identity

Trigonometric identities are equations that are true for all values of the variables involved, and they are immensely helpful in simplifying expressions and solving equations.
In this exercise, the identity used is the fundamental trigonometric identity:

• \(\sin^2 t + \cos^2 t = 1\)

This allows us to combine the parametric expressions \(x = 2 \sin t\) and \(y = 2 \cos t\) into a single equation by squaring both sides:

• \(x^2 = 4 \sin^2 t\)
• \(y^2 = 4 \cos^2 t\)

Adding these equations gives:

• \(\frac{x^2}{4} + \frac{y^2}{4} = \sin^2 t + \cos^2 t = 1\)

Thus, we arrive at a rectangular coordinate equation representing our semicircle. Having a good grasp of trigonometric identities helps in understanding the transformation of equations across different forms, simplifying the problem-solving process.