Explores the standard parametric forms for circles and ellipses, emphasizing the use of the Pythagorean trigonometric identity to eliminate the parameter.
What is the key trigonometric identity used to eliminate the parameter from equations involving sine and cosine?
The Pythagorean identity: \(\cos^2(t) + \sin^2(t) = 1\). This is fundamental for converting parametric forms of circles and ellipses into rectangular form.
What are the standard parametric equations for a circle of radius \(r\) centered at the origin \((0, 0)\)?
The standard equations are \(x = r\cos(t)\) and \(y = r\sin(t)\), for \(0 \le t \le 2\pi\).
Eliminate the parameter from \(x = 5\cos(t)\) and \(y = 5\sin(t)\).
From the equations, \(\cos(t) = \frac{x}{5}\) and \(\sin(t) = \frac{y}{5}\). Substitute into \(\cos^2(t) + \sin^2(t) = 1\) to get \((\frac{x}{5})^2 + (\frac{y}{5})^2 = 1\), which simplifies to \(x^2 + y^2 = 25\).
How do you parameterize a circle with radius \(r\) centered at a point \((h, k)\)?
The equations are \(x = h + r\cos(t)\) and \(y = k + r\sin(t)\). The \(h\) and \(k\) values provide the horizontal and vertical shifts from the origin.
Write the parametric equations for a circle with center \((-2, 3)\) and radius \(4\).
Using the form \(x = h + r\cos(t)\) and \(y = k + r\sin(t)\), we get \(x = -2 + 4\cos(t)\) and \(y = 3 + 4\sin(t)\).
What are the standard parametric equations for an ellipse centered at the origin, with a horizontal semi-axis of length \(a\) and a vertical semi-axis of length \(b\)?
The standard equations are \(x = a\cos(t)\) and \(y = b\sin(t)\), for \(0 \le t \le 2\pi\).
Eliminate the parameter from \(x = 3\cos(t)\) and \(y = 7\sin(t)\).
From the equations, \(\cos(t) = \frac{x}{3}\) and \(\sin(t) = \frac{y}{7}\). Substitute into \(\cos^2(t) + \sin^2(t) = 1\) to get \((\frac{x}{3})^2 + (\frac{y}{7})^2 = 1\), which is the equation of an ellipse.
Describe the curve given by \(x = 1 + 6\cos(t)\) and \(y = -4 + 2\sin(t)\).
This is an ellipse centered at \((1, -4)\). It has a horizontal semi-axis of length \(6\) and a vertical semi-axis of length \(2\).
How does swapping sine and cosine in the standard parameterization of a circle, i.e., \(x = r\sin(t), y = r\cos(t)\), affect the curve?
It still traces the same circle, but it changes the starting point and orientation. At \(t=0\), \((x,y) = (0,r)\) instead of \((r,0)\). As \(t\) increases, the curve is traced clockwise instead of counter-clockwise.
What range of values for the parameter \(t\) is typically required to trace a full circle or ellipse exactly once?
An interval of length \(2\pi\) is required. The standard interval is \([0, 2\pi]\) or \([-\pi, \pi]\).
What curve is traced by \(x = r\cos(t)\) and \(y = r\sin(t)\) for \(0 \le t \le \pi\)?
This traces the upper half of a circle of radius \(r\), starting at \((r, 0)\) and ending at \((-r, 0)\).
What is the orientation (direction) of the curve \(x = a\cos(t), y = b\sin(t)\) for \(a, b > 0\) as \(t\) increases from \(0\)?
Counter-clockwise. At \(t=0\), the point is \((a,0)\). At \(t=\pi/2\), the point is \((0,b)\). The curve moves from the positive x-axis towards the positive y-axis.
What curve is described by \(x = \sec(t)\) and \(y = \tan(t)\)?
This describes a hyperbola. It relies on the identity \(\sec^2(t) - \tan^2(t) = 1\). Substituting gives \(x^2 - y^2 = 1\).
How can you tell if the major axis of an ellipse, given as \(x=a\cos(t)\) and \(y=b\sin(t)\), is horizontal or vertical?
By comparing \(a\) and \(b\). If \(|a| > |b|\), the major axis is horizontal. If \(|b| > |a|\), the major axis is vertical.
Parameterize the ellipse given by the equation \(\frac{(x-2)^2}{9} + \frac{(y+1)^2}{16} = 1\).
Center is \((2, -1)\). Horizontal semi-axis \(a=\sqrt{9}=3\). Vertical semi-axis \(b=\sqrt{16}=4\). The equations are \(x = 2 + 3\cos(t)\) and \(y = -1 + 4\sin(t)\).
