Problem 27 - Algebra and Trigonometry

Question

Eliminate the parameter t. Then use the rectangular equation to sketch the plane curve represented by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of t. (If an interval for t is not specified, assume that \(-\infty< t <\infty.\)) \(x=2 \sin t, y=2 \cos t ; 0 \leq t<2 \pi\)

Step-by-Step Solution

Answer

The rectangular equation after eliminating the parameter \( t \) is \( x^2 + y^2 = 4 \) and it represents a circle with center at the origin and radius of 2. The curve is oriented in an anticlockwise direction.

Step 1: Eliminate Parameter \( t \)

In order to eliminate the parameter \( t \), the first step involves the Pythagorean Trigonometric Identity \( \sin^2 (t) + \cos^2 (t) = 1 \) will be used. Dividing \( x \) by \( 2 \) and \( y \) by \( 2 \) in the given equations gives: \( \sin(t) = x/2 \) and \( \cos(t) = y/2 \). Squaring these to get a common variable t, will provide: \( \sin^2(t) = (x^2)/4 \) and \( \cos^2(t) = (y^2)/4 \). Now, plug \( \sin^2(t) \) and \( \cos^2(t) \) into the Pythagorean Trigonometric Identity: \( (x^2)/4 + (y^2)/4 = 1 \). After simplification, \( x^2 + y^2 = 4 \) is obtained. This is the desired rectangular equation.

Step 2: Plot Equation

The equation \( x^2 + y^2 = 4 \) represents a circle centered at the origin with a radius of 2. Plot the circle and draw arrows in the direction of increasing \( t \), starting at \( t = 0 \), that is from the point (0, 2) and goes anticlockwise due to the properties of sine and cosine functions.

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental concept in trigonometry. It states that for any angle \( t \), the equation \( \sin^2(t) + \cos^2(t) = 1 \) holds true. This identity is derived from the Pythagorean theorem and relates the squares of the sine and cosine of an angle. To intuitively understand this, think about a right triangle where the hypotenuse is 1 (as in the unit circle), and \( \,\sin(t) \) and \( \,\cos(t) \) are the lengths of the other two sides.

• It is used to connect trigonometric functions with geometric concepts.
• Helps in transforming trigonometric functions from one to another.

When solving the exercise at hand, this identity assists in eliminating the parameter, \( t \), creating a seamless link between parametric equations and their rectangular equivalents.

Eliminate Parameter

The process of eliminating the parameter in a set of parametric equations involves deriving an equation without considering the parameter (in this case, \( t \)). For the given parametric equations, \( x = 2 \sin(t) \) and \( y = 2 \cos(t) \), eliminating \( t \) is possible by expressing \( \sin(t) \) and \( \cos(t) \) in terms of \( x \) and \( y \). First, divide both equations by 2:

• \( \sin(t) = x/2 \)
• \( \cos(t) = y/2 \)

After squaring these expressions, substitute them into the Pythagorean Identity:

• \( (x/2)^2 + (y/2)^2 = 1 \)

Simplifying gives the rectangular equation \( x^2 + y^2 = 4 \). This represents the original curve in a form that doesn't involve the parameter \( t \). This transformation is critical for visualizing and further analyzing the behavior of the curve.

Rectangular Equation

A rectangular equation is an algebraic equation involving only the Cartesian coordinates \( x \) and \( y \), and it expresses a curve in the plane. Once a parametric form is converted to a rectangular equation, it often becomes easier to recognize and plot. For example, in this exercise, eliminating \( t \) yielded the rectangular equation \( x^2 + y^2 = 4 \). This commutatively represents a circle centered at the origin with radius 2.

• Benefits: Makes plotting simpler and helps understand the curve's overall behavior.
• Comparison: While parametric descriptions can handle more complex cases like spirals or oscillations, rectangular equations benefit quick interpretation.

Thus, understanding how to derive rectangular equations broadens the scope of analysis one can apply to various mathematical curves.

Plotting Plane Curves

Plotting plane curves is essential for visualizing mathematical relationships expressed by equations. In the context of this exercise, plotting the curve defined by \( x^2 + y^2 = 4 \) gives us a circle of radius 2 centered at the origin.To correctly plot the curve:

• Identify all intercepts, symmetries, and specific points such as the top \((0, 2)\), bottom \((0, -2)\), right \((2, 0)\), and left \((-2, 0)\).
• Understand the increasing direction of the parameter \( t \). Starting at \( t = 0 \) (\( (0, 2) \), moving counterclockwise due to how \( \sin(t) \) and \( \cos(t) \) evolve).

Using these steps, correctly gauge the orientation of the curve, which is particularly useful for physics-based problems where direction (e.g., in vector fields) matters. Each plotted curve provides insights into how mathematical relationships behave visually.