Problem 19
Question
A pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular-coordinate equation for the curve by eliminating the parameter. $$x=\sec t, \quad y=\tan t, \quad 0 \leq t<\pi / 2$$
Step-by-Step Solution
Verified Answer
(a) Sketch starts at (1,0) going to \(+\infty\); (b) Rectangular equation: \(x^2 = 1 + y^2\) with \(x \geq 1\).
1Step 1: Understanding the Parametric Equations
The given parametric equations are \( x = \sec t \) and \( y = \tan t \) with the parameter \( t \) in the interval \([0, \pi/2)\). Here, \( \sec t = \frac{1}{\cos t} \) and \( \tan t = \frac{\sin t}{\cos t} \).
2Step 2: Sketching the Curve
To sketch the curve, consider values of \( t \) within its interval. For example, when \( t = 0 \), \( x = \sec(0) = 1 \) and \( y = \tan(0) = 0 \). As \( t \) increases towards \( \pi/2 \), \( \sec t \) and \( \tan t \) both approach infinity. This suggests the curve emanates from (1,0) and goes towards infinity along the positive y-direction.
3Step 3: Eliminate the Parameter to Find a Rectangular Equation
To eliminate the parameter \( t \), use the Pythagorean identity: \( \sec^2 t = 1 + \tan^2 t \). Substitute \( x = \sec t \) and \( y = \tan t \), then \( x^2 = \sec^2 t = 1 + \tan^2 t = 1 + y^2 \). Thus, the rectangular-coordinate equation is \( x^2 = 1 + y^2 \).
4Step 4: Verify the Solution Domain
While \( x^2 = 1 + y^2 \) typically represents a hyperbola, due to the constraint \( 0 \leq t < \pi/2 \), \( x = \sec t \geq 1 \) and \( y = \tan t \geq 0 \). Therefore, the valid portion of the curve is for \( x \geq 1 \) on the hyperbola. This confirms the curve begins at (1,0) and goes upwards.
Key Concepts
Understanding Rectangular-Coordinate EquationsExploring the Pythagorean IdentityBasics of Curve Sketching
Understanding Rectangular-Coordinate Equations
Rectangular-coordinate equations, sometimes known as Cartesian equations, represent curves in the plane using two variables, commonly labeled as \( x \) and \( y \). These equations help to plot points on a graph, showing how these two variables relate to each other.
In the context of the exercise, a rectangular-coordinate equation was derived from parametric equations. Parametric equations, like \( x = \sec t \) and \( y = \tan t \), use a third variable, often \( t \), to parameterize the relationship between \( x \) and \( y \).
By eliminating the parameter \( t \), as demonstrated in the step-by-step solution, you obtain a rectangular equation expressing \( x \) directly in terms of \( y \), or vice versa. This was achieved using trigonometric identities, transforming the parameters into a coherent equation \( x^2 = 1 + y^2 \).
In the context of the exercise, a rectangular-coordinate equation was derived from parametric equations. Parametric equations, like \( x = \sec t \) and \( y = \tan t \), use a third variable, often \( t \), to parameterize the relationship between \( x \) and \( y \).
By eliminating the parameter \( t \), as demonstrated in the step-by-step solution, you obtain a rectangular equation expressing \( x \) directly in terms of \( y \), or vice versa. This was achieved using trigonometric identities, transforming the parameters into a coherent equation \( x^2 = 1 + y^2 \).
- Parametric elimination provides a more straightforward format to sketch curves.
- Helps in identifying relationships and constraints on variables.
- Essential in simplifying complex parametric curves.
Exploring the Pythagorean Identity
The Pythagorean identity is a fundamental concept in trigonometry. It states that \( \sec^2 t = 1 + \tan^2 t \), a powerful equation derived from the basic identity \( \sin^2 t + \cos^2 t = 1 \).
This identity is invaluable when dealing with parametric equations involving secant and tangent. In this exercise, it played a central role in eliminating the parameter \( t \) to find the rectangular equation. By substituting \( \sec t = x \) and \( \tan t = y \), the equation \( \sec^2 t = 1 + \tan^2 t \) directly translates into \( x^2 = 1 + y^2 \). This step is crucial for simplifying the description of a curve defined by trigonometric functions.
The Pythagorean identity’s application here shows how identities can facilitate the transition from a set of parametric equations to a single rectangular equation, thus making it easier to analyze and graph the curve.
This identity is invaluable when dealing with parametric equations involving secant and tangent. In this exercise, it played a central role in eliminating the parameter \( t \) to find the rectangular equation. By substituting \( \sec t = x \) and \( \tan t = y \), the equation \( \sec^2 t = 1 + \tan^2 t \) directly translates into \( x^2 = 1 + y^2 \). This step is crucial for simplifying the description of a curve defined by trigonometric functions.
The Pythagorean identity’s application here shows how identities can facilitate the transition from a set of parametric equations to a single rectangular equation, thus making it easier to analyze and graph the curve.
Basics of Curve Sketching
Curve sketching involves drawing the graph of an equation or set of equations. It's a visual representation that provides insight into the behavior of mathematical functions.
In this exercise, the process began with parameter values, starting when\( t = 0 \): \( x = \sec(0) = 1 \) and \( y = \tan(0) = 0 \). As \( t \) approached \( \pi/2 \), both \( \sec t \) and \( \tan t \) increased towards infinity. This indicates that the curve begins at the point \( (1,0) \) and extends infinitely in the positive y-direction.
A critical aspect of sketching curves from parametric equations is understanding the domain restrictions. For these equations, while typically a hyperbola, the domain \( 0 \leq t < \pi/2 \) limits\( x \geq 1 \) and \( y \geq 0 \).
Sketching such curves entails:
In this exercise, the process began with parameter values, starting when\( t = 0 \): \( x = \sec(0) = 1 \) and \( y = \tan(0) = 0 \). As \( t \) approached \( \pi/2 \), both \( \sec t \) and \( \tan t \) increased towards infinity. This indicates that the curve begins at the point \( (1,0) \) and extends infinitely in the positive y-direction.
A critical aspect of sketching curves from parametric equations is understanding the domain restrictions. For these equations, while typically a hyperbola, the domain \( 0 \leq t < \pi/2 \) limits\( x \geq 1 \) and \( y \geq 0 \).
Sketching such curves entails:
- Identifying starting and ending points.
- Recognizing the direction of the curve.
- Understanding asymptotic behavior as \( t \) increases.
Other exercises in this chapter
Problem 18
Sketch a graph of the polar equation, and express the equation in rectangular coordinates. $$r=-1$$
View solution Problem 19
Sketch \(z_{1}, z_{2}, z_{1}+z_{2},\) and \(z_{1} z_{2}\) on the same complex plane. $$z_{1}=2-i, \quad z_{2}=2+i$$
View solution Problem 20
Sketch \(z_{1}, z_{2}, z_{1}+z_{2},\) and \(z_{1} z_{2}\) on the same complex plane. $$z_{1}=-1+i, \quad z_{2}=2-3 i$$
View solution Problem 20
A pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular-coordinate equation for the curve
View solution