Problem 22 - Calculus Early Transcendentals

Question

Eliminate the parameter to find a description of the following circles or circular arcs in terms of \(x\) and \(y .\) Give the center and radius, and indicate the positive orientation. $$x=3 \cos t, y=3 \sin t ; 0 \leq t \leq \pi / 2$$

Step-by-Step Solution

Answer

Answer: The equation of the circle is \(x^2 + y^2 = 9\), with a center at \((0, 0)\) and a radius of \(3\).

Step 1: Use the trigonometric identity

We are given the parametric equations: $$x = 3 \cos t, \quad y = 3 \sin t$$ To eliminate the parameter \(t\), square both equations and add them together: $$x^2 = 9\cos^2 t \quad \text{and} \quad y^2 = 9\sin^2 t$$ Then, we obtain $$x^2 + y^2 = 9(\cos^2 t + \sin^2 t)$$

Step 2: Apply the identity and simplify

Using the given trigonometric identity: $$x^2 + y^2 = 9(\cos^2 t + \sin^2 t) = 9(1)$$ Simplifying the equation, we get: $$x^2 + y^2 = 9$$

Step 3: Find the center and radius of the circle

The equation we found above is the standard equation of a circle with center \((h, k)\) and radius \(r\), which means the equation is in the form $$(x-h)^2 + (y-k)^2 = r^2$$ Comparing the two equations, we get: $$x^2 + y^2 = 9 \Rightarrow (x-0)^2 + (y-0)^2 = 3^2$$ So, the center of the circle is \((0, 0)\), and the radius \(r\) is \(3\).

Step 4: Determine the orientation and range of the circle

Since \(0 \leq t \leq \pi / 2\), we know that the circle's arc is within the first quadrant. Therefore, it covers a quarter of a full circle with a positive (counterclockwise) orientation, as \(\sin\) and \(\cos\) are both non-negative in the first quadrant. In conclusion, we have the equation of a circle with a center at \((0, 0)\), a radius of \(3\), and a positive orientation that covers the first quadrant: $$x^2 + y^2 = 9$$

Key Concepts

Trigonometric Identity

Understanding trigonometric identities is crucial when working with parametric equations, especially those representing circles. A fundamental identity used in the analysis of circles is the Pythagorean trigonometric identity, which states that for any angle \(t\):\[ \cos^2 t + \sin^2 t = 1 \]This identity demonstrates the inherent relationship between the cosine and sine of an angle—a cornerstone concept in trigonometry that arises from the definition of these functions on the unit circle. When dealing with parametric equations that describe a circle, such as \(x = r\cos t\) and \(y = r\sin t\), where \(r\) is the radius, squaring both and adding them—as executed in the exercise—will always leverage this identity to simplify the expression, ultimately leading to the standard equation of a circle.
Once you familiarize yourself with this trigonometric identity, you'll find it much easier to manage equations involving sine and cosine, allowing for a more streamline process when eliminating parameters or proving other advanced concepts.

Eliminate the Parameter

Eliminating the parameter in parametric equations is a method of transforming these equations into their standard form. For a circle, the given parametric equations are typically a pair of functions for \(x\) and \(y\), both dependent on a parameter \(t\). To eliminate the parameter, we use algebraic manipulation and trigonometric identities to rewrite the equations without \(t\), which will yield an equation only in terms of \(x\) and \(y\).
As seen in our exercise, squaring both parametric formulas and utilizing the Pythagorean trigonometric identity allowed us to eliminate the parameter \(t\). This process isolates \(x\) and \(y\) within a familiar equation representing a geometrical figure—in this case, a circle. Mastering this skill is not only pivotal in geometry but also in higher mathematics, where parametric equations are commonly encountered.

Standard Equation of a Circle

The standard equation of a circle is a straightforward way to represent a circle algebraically. It is expressed as:\[ (x - h)^2 + (y - k)^2 = r^2 \]Where \( (h, k) \) is the center of the circle and \(r\) is its radius. After eliminating the parameter, the resulting equation can be compared to this standard form to identify the circle's properties. In our case, the equation \(x^2 + y^2 = 9\) equates to \( (x - 0)^2 + (y - 0)^2 = 3^2 \) where you can clearly see the center at \( (0, 0) \) and the radius \(3\).
This simple yet powerful equation encapsulates all the information needed to describe a circle fully. Once you understand this, you can analyze the circle's position and size, which are fundamental in both pure and applied mathematics.

Circle Orientation and Range

Circle orientation refers to the direction in which the circle is traversed, and range describes the extent of the circle covered by the parametric equations. For our exercise, the orientation is positive when the parameter \(t\) increases in the interval \(0 \leq t \leq \pi/2\). A positive orientation means counterclockwise movement, which is often considered the 'standard' direction in mathematics.
The range, given by the interval for \(t\), restricts our circle to a circular arc rather than a complete circle. Since the interval is from \(0\) to \(\pi/2\), it tells us that the arc only spans the first quadrant of the coordinate plane where both \(x\) and \(y\) are positive, representing a quarter of a full circle. Recognizing the orientation and range is necessary for properly conceptualizing the circle's segment being described and is invaluable when solving related real-world problems, such as determining the path of an object moving in a circular trajectory.