Question

Consider the polar curve \(r=2 \sec \theta\). a. Graph the curve on the intervals \((\pi / 2,3 \pi / 2),(3 \pi / 2,5 \pi / 2)\) and \((5 \pi / 2,7 \pi / 2) .\) In each case, state the direction in which the curve is generated as \(\theta\) increases. b. Show that on any interval \((n \pi / 2,(n+2) \pi / 2),\) where \(n\) is an odd integer, the graph is the vertical line \(x=2\).

Short answer

In this problem, we were given a polar curve \(r = 2\sec\theta\), and we had to graph it for three given intervals of \(\theta\), while also proving that the graph is always a vertical line \(x=2\) within any odd integer intervals of \(\theta\). After converting the polar equation to the rectangular equation, we found that \(x = 2\), which is a vertical line passing through \((2,0)\). We plotted the curve for the given intervals and observed the direction of the curve generated as \(\theta\) increases. Furthermore, we concluded that on any interval \((n\pi/2,(n+2)\pi/2)\), where \(n\) is an odd integer, the graph is always the vertical line \(x = 2\), because the value of \(x\) remains constant for any value of \(\theta\).

Step-by-step solution

Convert the Polar Equation to a Rectangular Equation

We have the polar equation \(r = 2\sec\theta\). Recall the trigonometric identity \(\sec\theta = \frac{1}{\cos\theta}\). So we have \(r = \frac{2}{\cos\theta}\). We can now substitute \(x = r\cos\theta\) and \(y = r\sin\theta\) to get the rectangular equation. Using the polar to rectangular conversion formulas, we have: \(x = r\cos\theta = \frac{2}{\cos\theta}\cos\theta = 2.\) Hence, the rectangular equation is \(x = 2.\) This is a vertical line passing through point \((2,0)\).

Graph the Curve for Given Intervals of \(\theta\)

To graph the curve for the given intervals of \(\theta\), we need to consider the direction in which the curve is generated as \(\theta\) increases. For \((\pi/2, 3\pi/2)\): As \(\theta\) increases from \(\pi/2\) to \(3\pi/2\), \(x\) remains constant at \(x = 2\). The direction of the curve is generated from the top (positive y-axis) to the bottom (negative y-axis) vertically along the line \(x = 2\). For \((3\pi/2, 5\pi/2)\): As \(\theta\) increases from \(3\pi/2\) to \(5\pi/2\), \(x\) remains constant at \(x = 2\). The direction of the curve is generated from the bottom (negative y-axis) to the top (positive y-axis) vertically along the line \(x = 2\). For \((5\pi/2, 7\pi/2)\): As \(\theta\) increases from \(5\pi/2\) to \(7\pi/2\), \(x\) remains constant at \(x = 2\). The direction of the curve is generated from the top (positive y-axis) to the bottom (negative y-axis) vertically along the line \(x = 2\).

Prove that the Graph is Always the Vertical Line \(x=2\) on Odd Intervals of \(\theta\)

As we know that the rectangular equation is \(x = 2\) which is a vertical line passing through \((2,0)\). We need to prove that for any interval \((n\pi/2,(n+2)\pi/2),\) where \(n\) is an odd integer, the graph is the same vertical line \(x = 2\). Recall the equation, \(x = 2.\) Here, x does not depend on the value of \(\theta\). Therefore, as \(\theta\) increases in any odd integer intervals such as \((\pi/2, 3\pi/2),\) \((3\pi/2, 5\pi/2),\) and \((5\pi/2, 7\pi/2)\), \(x\) always remains constant at \(x = 2\). Hence, we can conclude that on any odd integer interval \((n\pi/2,(n+2)\pi/2)\), the graph is always the vertical line \(x = 2\).

Key concepts

Polar to Rectangular Conversion

Converting polar equations to rectangular form is a fundamental skill in calculus that bridges the gap between different coordinate systems. The process involves using the Cartesian-coordinates relationship with polar coordinates, where any polar point \( (r, \theta) \) corresponds to the Cartesian point \( (x, y) \) through the conversions \( x = r\cos(\theta) \) and \( y = r\sin(\theta) \).

In the example of the polar equation \( r = 2\sec(\theta) \), we utilize the properties of the trigonometric identity \( \sec(\theta) = \frac{1}{\cos(\theta)} \) to find the corresponding rectangular representation. Since \( \sec(\theta) \) is the reciprocal of \( \cos(\theta) \) and \( x = r\cos(\theta) \), it naturally simplifies to \( x = 2 \) without the dependence on \( \theta \), leading us to a vertical line in the Cartesian coordinate system. This demonstrates the implications of a polar equation that does not explicitly depend on \( \theta \) when converted to rectangular form – it results in the simplicity of a line equation.

Graphing Polar Equations

Graphing polar equations can be nuanced, as the visual representation of these equations is fundamentally different from that of Cartesian graphs. Polar graphs revolve around a central point, the pole, and they extend in radial directions determined by the angle \( \theta \), while radii depend on the function \( r(\theta) \).

When graphing polar equations like \( r = 2\sec(\theta) \), one useful step is to convert to rectangular coordinates to better understand the structure of the graph. In the steps outlined, due to the nature of the \( \sec(\theta) \) function, the graph represents a vertical line at \( x = 2 \) for all specified intervals of \( \theta \), illustrating how no matter the angle, all points on this polar curve have the same \( x \) coordinate. It's important when graphing such curves to pay attention to the range of \( \theta \) as it dictates the directionality and extent of the curve on the graph.

Trigonometric Identities

Trigonometric identities form the bedrock of simplifying expressions in calculus that involve angles. They encapsulate the fundamental relationships between the six trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent. One of the most widely used trigonometric identities is \( \sec(\theta) = \frac{1}{\cos(\theta)} \), showcased in our example problem.

Understanding identities like the reciprocal one between secant and cosine allows us to manipulate and simplify polar equations for easier conversion to rectangular form. These identities also contribute to the determination of the symmetry, bounds, and behavior of trigonometric functions as they apply to both polar and rectangular equations. The precise application of these identities is crucial when solving complex calculus problems, illustrating why securing a strong grasp on them is essential for students.