Problem 14 - College Trigonometry

Question

In Exercises \(1-20\), plot the set of parametric equations by hand. Be sure to indicate the orientation imparted on the curve by the parametrization. $$ \left\\{\begin{array}{l} x=2 \tan (t) \\ y=\cot (t) \end{array} \text { for } 0

Step-by-Step Solution

Answer

The curve of the parametric equations \(x = 2 \tan(t)\), \(y = \cot(t)\) for \(0

Step 1: Understand the Parametric Equations

The given parametric equations are: \( x = 2 \tan(t) \) and \( y = \cot(t) \). These equations are functions of a parameter \( t \) where \( 0 < t < \frac{\pi}{2} \). Our task is to plot these equations on the coordinate plane and determine the orientation.

Step 2: Determine the Range of Parameters

Since \( 0 < t < \frac{\pi}{2} \), we must determine the behavior of \( x \) and \( y \) as \( t \) approaches these limits:- As \( t \to 0^+ \), \( \tan(t) \to 0 \) and \( \cot(t) \to \infty \).- As \( t \to \frac{\pi}{2}^- \), \( \tan(t) \to \infty \) and \( \cot(t) \to 0 \).This helps to establish the direction in which the curve will be drawn. The curve starts from \((0, \infty)\) and moves towards \((\infty, 0)\).

Step 3: Transform to Cartesian Coordinates

To better understand the relationship between \( x \) and \( y \), eliminate the parameter \( t \) by expressing \( \tan(t) \) and \( \cot(t) \) in terms of each other:\[ \tan(t) = \frac{1}{\cot(t)} \]Thus, substitute \( \tan(t) \) into the equation for \( x \):\[ x = 2 \left(\frac{1}{\cot(t)}\right) = \frac{2}{y} \]Rearrange to find the Cartesian relationship between \( x \) and \( y \):\[ xy = 2 \]

Step 4: Sketch the Graph Based on Orientation

The Cartesian equation \( xy = 2 \) represents a rectangular hyperbola centered at the origin. To plot:1. Identify points using the parametric equations, for example: - If \( t = \frac{\pi}{4} \), then \( x = 2 \times 1 = 2 \) and \( y = 1 \), giving the point \((2,1)\).2. Use the Cartesian equation that confirms the points and direction.3. The orientation is from top-left to bottom-right and shows that the curve moves from \((0, \infty)\) to \((\infty, 0)\) in the first quadrant.

Key Concepts

Rectangular Hyperbola

A rectangular hyperbola is a special type of hyperbola where the asymptotes are perpendicular to each other, intersecting at right angles. In our specific problem, the hyperbola is represented by the Cartesian equation \(xy = 2\). This form is central to the shape of a rectangular hyperbola.

• Understanding Orientation and Shape
  The hyperbola has two distinct branches, which splay out towards infinity in opposite quadrants. Our rectangle is centered around the axes intersection, having symmetry about these lines.
  The graph will be symmetrical in two quadrants, but in the case of our problem, it focuses on the first quadrant.
  
• Graph Features
  You'll notice that as \(x\) goes to infinity, \(y\) approaches zero and vice versa. The hyperbola does not cross its asymptotes, showing important characteristics in both Cartesian and parametric plots.

The equation \(xy = 2\) shows how \(x\) and \(y\) are inversely related, maintaining a constant product value. It's an interesting property of rectangular hyperbolas that visualization through specified ranges makes more evident.

Cartesian Coordinates

Cartesian coordinates are a system that specifies each point uniquely in a plane using a pair of numerical coordinates. These are distances to two perpendicular-directed lines, which is the x-axis and the y-axis.

• Transforming Parametric Equations
  Our task involves transforming parametric equations into a Cartesian form, to understand more about the curves and their interrelations. Starting with the equations \(x = 2 \tan(t)\) and \(y = \cot(t)\), we eliminated the parameter \(t\) to get \(xy = 2\).
  The result is significant because it shows the relationship independent of \(t\) and places it in a framework many are familiar with, the Cartesian plane.
• X and Y Axis Interactions
  Understanding how \(x\) and \(y\) interact helps in predicting the values and the plot direction. In our example, the relationship dictates that an increase in \(x\) results in a proportional decrease in \(y\), and vice versa. This inverse proportionality is essential for graphing and mathematical representation.

Trigonometric Functions

Trigonometric functions are some of the most fundamental tools in mathematics, deeply connected with angles and ratios, in various applications from graphs to waves.

• Tangent and Cotangent Functions
  In the given parametric equations, \(x = 2 \tan(t)\) uses the tangent function, which relates to similar triangles and circular functions. \(y = \cot(t)\) makes use of the cotangent function, which is the reciprocal of the tangent. They are thus complementary, showcasing their interplay over the interval where \(0 < t < \frac{\pi}{2}\).
  These functions help translate an angle \(t\) into a meaningful curve when plotted onto the Cartesian system. Each value of \(t\) gives a point that collectively forms the curve.
• Behavior within Allowed Range
  As \(t\) nears the limit \(0\), \(tan(t)\rightarrow 0\) and \(cot(t)\to \infty\), indicating the curve starts at infinity on the y-axis. At approximately \(t = \frac{\pi}{2}\), \(tan(t)\rightarrow \infty\) and \(cot(t)\rightarrow 0\), reflecting the trajectory towards infinity on the x-axis.
  This behavior allows us to understand the progressive motion and orientation of our curve, starting near and eventually moving apart as limits close in.

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Problem 13

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Problem 14

Other exercises in this chapter

Problem 13

Solve for the remaining side(s) and angle(s), if possible, using any appropriate technique. $$ a=16, \alpha=63^{\circ}, b=20 $$

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Problem 13

Plot the point given in polar coordinates and then give three different expressions for the point such that (a) \(r0\) and \(\theta \leq 0\) (c) \(r>0\) and \(\

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Problem 14

Find the component form of the vector \(\vec{v}\) using the information given about its magnitude and direction. Give exact values. \(\|\vec{v}\|=12\); when dra

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Problem 14

In Exercises \(1-20\), plot the graph of the polar equation by hand. Carefully label your graphs. Limaçon: \(r=1-2 \sin (\theta)\)

View solution

Key Concepts

Rectangular Hyperbola

A rectangular hyperbola is a special type of hyperbola where the asymptotes are perpendicular to each other, intersecting at right angles. In our specific problem, the hyperbola is represented by the Cartesian equation \(xy = 2\). This form is central to the shape of a rectangular hyperbola.

• Understanding Orientation and Shape
  The hyperbola has two distinct branches, which splay out towards infinity in opposite quadrants. Our rectangle is centered around the axes intersection, having symmetry about these lines.
  The graph will be symmetrical in two quadrants, but in the case of our problem, it focuses on the first quadrant.
  
• Graph Features
  You'll notice that as \(x\) goes to infinity, \(y\) approaches zero and vice versa. The hyperbola does not cross its asymptotes, showing important characteristics in both Cartesian and parametric plots.

The equation \(xy = 2\) shows how \(x\) and \(y\) are inversely related, maintaining a constant product value. It's an interesting property of rectangular hyperbolas that visualization through specified ranges makes more evident.

Cartesian Coordinates

Cartesian coordinates are a system that specifies each point uniquely in a plane using a pair of numerical coordinates. These are distances to two perpendicular-directed lines, which is the x-axis and the y-axis.

• Transforming Parametric Equations
  Our task involves transforming parametric equations into a Cartesian form, to understand more about the curves and their interrelations. Starting with the equations \(x = 2 \tan(t)\) and \(y = \cot(t)\), we eliminated the parameter \(t\) to get \(xy = 2\).
  The result is significant because it shows the relationship independent of \(t\) and places it in a framework many are familiar with, the Cartesian plane.
• X and Y Axis Interactions
  Understanding how \(x\) and \(y\) interact helps in predicting the values and the plot direction. In our example, the relationship dictates that an increase in \(x\) results in a proportional decrease in \(y\), and vice versa. This inverse proportionality is essential for graphing and mathematical representation.

Trigonometric Functions

Trigonometric functions are some of the most fundamental tools in mathematics, deeply connected with angles and ratios, in various applications from graphs to waves.

• Tangent and Cotangent Functions
  In the given parametric equations, \(x = 2 \tan(t)\) uses the tangent function, which relates to similar triangles and circular functions. \(y = \cot(t)\) makes use of the cotangent function, which is the reciprocal of the tangent. They are thus complementary, showcasing their interplay over the interval where \(0 < t < \frac{\pi}{2}\).
  These functions help translate an angle \(t\) into a meaningful curve when plotted onto the Cartesian system. Each value of \(t\) gives a point that collectively forms the curve.
• Behavior within Allowed Range
  As \(t\) nears the limit \(0\), \(tan(t)\rightarrow 0\) and \(cot(t)\to \infty\), indicating the curve starts at infinity on the y-axis. At approximately \(t = \frac{\pi}{2}\), \(tan(t)\rightarrow \infty\) and \(cot(t)\rightarrow 0\), reflecting the trajectory towards infinity on the x-axis.
  This behavior allows us to understand the progressive motion and orientation of our curve, starting near and eventually moving apart as limits close in.