Question

You may use a graphing calculator to solve the following problems. True or False The graph of the parametric curve \(x=3 \cos t$$y=4 \sin t\) is a circle. Justify your answer.

Short answer

False. The graph of the given parametric equations is not a circle as the coefficients of the cosine and sine functions are not equal, which are necessary for a circle in the parametric form.

Step-by-step solution

Familiarize with the Equation of a Circle

Remind yourself that in parametric form, the equation of a circle is typically either \(x=r\cos(t)\) and \(y=r\sin(t)\) or \(x=h+r\cos(t)\) and \(y=k+r\sin(t)\) where \(r\) is the radius of the circle and \((h,k)\) are the coordinates of the center.

Analyze given parametric Equations

Looking at the given parametric equations, \(x = 3\cos(t)\) and \(y = 4\sin(t)\), you notice that the coefficient of the cosine function (which is 3) is different from the coefficient of the sine function (which is 4). If this were a circle, the coefficients, representing the radius, would be equal.

Draw Conclusion

Because the coefficients are different, the graph for these parametric equations is not a circle. Hence, the answer is False.

Key concepts

Graphing Calculator

A graphing calculator is an invaluable tool for visualizing mathematical concepts and solving complex problems. It allows students to input equations, including parametric ones, and see their graphical representations. Using a graphing calculator to plot the given parametric equations, \(x=3 \cos t\) and \(y=4 \sin t\), offers an immediate visual confirmation of whether these equations represent a circle. By inputting these equations, students can observe the shape and behavior of the curve. Through this technology, they can explore the relationship between the algebraic form of a parametric curve and its geometric representation. Additionally, the graphing calculator's ability to handle trigonometric functions makes it an essential tool for this exercise.

Parametric Curve

A parametric curve is a graph plotted using a set of parameterized equations, where the x and y coordinates are both defined in terms of a third variable, typically denoted as 't' or 'theta' for angles. Unlike the standard y=f(x) format, parametric equations allow us to represent more complex shapes, such as circles and ellipses, with ease. These curves are particularly useful when a relationship between two variables is best expressed in terms of another parameter. The given equations \(x=3 \cos t\) and \(y=4 \sin t\) define a parametric curve. As the parameter 't' varies, the equations trace out a path in the coordinate plane. To determine the nature of this path, we assess the relationship between the x and y components. If they correspond to a constant radius value 'r', we might be describing a circle.

Trigonometric Functions

Trigonometric functions, such as sine (\(\sin\)) and cosine (\(\cos\)), encapsulate the relationship between the angles and lengths of a right-angled triangle. They also extend beyond triangles to describe periodic phenomena such as waves. In the context of parametric equations for a circle, the sine and cosine functions represent the projection of a rotating radius onto the x and y-axes, respectively. This rotation generates the circle's circumference when the radius is consistent for both functions, leading to the traditional parametric equations for a circle \(x=r\cos(t)\) and \(y=r\sin(t)\). These functions are central to understanding why the original exercise's equations do not describe a circle: the radiuses represented by the coefficients (3 for cosine, 4 for sine) are inconsistent, indicating an elliptical shape rather than a circular one.

Radius of a Circle

In geometry, the radius of a circle is the distance from the center to any point on its circumference. It is constant for all points on the circle, which is a crucial aspect of a circle's definition. This constant radius, denoted by 'r', is central to the equations of a circle in both standard form \(x^2 + y^2 = r^2\) and parametric form \(x=r\cos(t)\), \(y=r\sin(t)\). The value of 'r' must be the same in both the x and y parametric equations to ensure that each point \((x,y)\) lies on the circumference of the same circle. In our initial problem, since the coefficients of the sine and cosine functions (acting as the radius) are different, the resulting shape does not have a constant radius from a central point and therefore cannot be a circle.