Problem 13 - Precalculus Mathematics for Calculus

Question

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

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

Step 1: Understanding the Parametric Equations

The given parametric equations are \(x = 2 \sin t\) and \(y = 2 \cos t\). The parameter \(t\) varies from 0 to \(\pi\). These equations describe how the coordinates \(x\) and \(y\) change as \(t\) changes.

Step 2: Sketching the Curve

The equations resemble the trigonometric identities for a circle: in a full parameter interval \(0 \leq t \leq 2\pi\), these would trace a circle of radius 2 centered at the origin. However, since \(t\) varies from 0 to \(\pi\), the curve represents a semicircle (the upper half of the circle) centered at the origin with radius 2.

Step 3: Eliminate the Parameter

To find a rectangular equation, we eliminate \(t\) by using the trigonometric identity \(\sin^2 t + \cos^2 t = 1\). Solving for \(\sin t\) and \(\cos t\) gives us \(\sin t = \frac{x}{2}\) and \(\cos t = \frac{y}{2}\). Substituting these into the identity, we get \((\frac{x}{2})^2 + (\frac{y}{2})^2 = 1\).

Step 4: Simplifying the Equation

Simplifying \((\frac{x}{2})^2 + (\frac{y}{2})^2 = 1\) by multiplying everything by 4 (the least common multiple of the denominators) yields \(x^2 + y^2 = 4\). This is the equation of a circle with radius 2 centered at the origin.

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in mathematics that relate the angles and sides of triangles in various ways. The most widely used identity, known as the Pythagorean identity, is \[ \sin^2 t + \cos^2 t = 1 \] This identity is crucial when dealing with parametric equations involving sine and cosine functions, as it enables us to convert parametric equations into rectangular form.

• Sine and Cosine: These are primary trigonometric functions used to represent circular motion.
• Sinusoidal Nature: The parameter \( t \) represents an angle, measured in radians. As \( t \) varies, it causes \( \sin t \) and \( \cos t \) to oscillate between -1 and 1.
• Utility: By expressing \( x \) and \( y \) in terms of \( \sin t \) and \( \cos t \), we can analyze curves described by parametric equations.

By substituting \( \sin t = \frac{x}{2} \) and \( \cos t = \frac{y}{2} \) into the Pythagorean identity, we effectively eliminate the parameter \( t \) and find a corresponding rectangular equation.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are used to position a point in a plane using two perpendicular lines, usually referred to as the x-axis and y-axis.

• Coordinates: Any point on the plane is defined by a pair of numbers (x, y), representing its position.
• Relation to Parametric Equations: Parametric equations use a parameter (in this case \( t \)) to express \( x \) and \( y \) with separate equations based on that parameter.
• Conversion: To convert parametric equations to the rectangular form, eliminate the parameter by finding relations between \( x \) and \( y \).

In our exercise, both \( x \) and \( y \) were expressed in terms of trigonometric functions of \( t \), allowing us to utilize identities to solve for a rectangular equation. This turned the parametric expression into a standard equation suitable for plotting or analyzing.

Circle Equation

The equation of a circle in rectangular form is one of the simplest and most recognizable forms in geometry. It is generally given by \[ x^2 + y^2 = r^2 \] where \( r \) is the radius of the circle.A circle with a specified center (h, k) can be represented as \[ (x - h)^2 + (y - k)^2 = r^2 \] In our example, the derived equation, \( x^2 + y^2 = 4 \), represents a circle of radius 2 centered at the origin (0, 0).

• Equation Characteristics: The simplicity of the equation \( x^2 + y^2 = r^2 \) makes it easy to identify and graph circles.
• Semicircle Insight: With the parameter \( t \) limited from 0 to \( \pi \), the curve represents only the top half (semicircle) of the complete circle.
• Graphical Representation: Such equations are used frequently to model circular shapes across various disciplines.

Understanding these foundational equations allows students to tackle more complex geometric problems with ease.