Problem 48 - Contemporary Precalculus

Question

(a) Find a parameterization of the line segment joining (-5,-3) and \((7,4),\) as in Exercises \(45-47\) (b) Explain why another parameterization of this line segment is given by $$\begin{array}{l}x=-5+12 \sin t \quad \text { and } \\\y=-3+7 \sin t \quad(0 \leq t \leq \pi / 2)\end{array}$$ (c) Use the trace feature to verify that the segment is traced out twice when the \(t\) -range in part (b) is changed to \(0 \leq t \leq \pi(\text { use } t-\text { step }=\pi / 20) .\) Explain why. (d) What happens when \(0 \leq t \leq 2 \pi ?\)

Step-by-Step Solution

Answer

Answer: The line segment is traced out four times when the t-range is changed to 0 ≤ t ≤ 2π.

Step 1: Parameterizing the line segment

To find a parameterization of the line segment joining the points (-5, -3) and (7, 4), we can use the formula: (x, y) = (1 - t)(-5, -3) + t(7, 4) This gives us the parameterized equations for x and y as: x = -5 + 12t y = -3 + 7t where 0 ≤ t ≤ 1.

Step 2: Explaining the given parameterization

The given parameterization is another correct representation of the line segment because, by using the properties of sine function, we can observe that: - When t = 0, \(\sin t = 0\), and thus, x = -5, and y = -3. - When t = π/2, \(\sin t = 1\), and thus, x = -5 + 12 = 7, and y = -3 + 7 = 4. As the sine function oscillates only between -1 and 1, only the given points will be obtained when the sine function is applied in the equations provided. Therefore, the given parameterization is another correct representation of the line segment.

Step 3: Verifying the segment is traced out twice

When the t-range in part (b) is changed to 0 ≤ t ≤ π, we can observe that the segment would be traced out twice because the sine function has a period of 2π, which means that it repeats itself every 2π. In the given range, we see that the sine function oscillates from 0 to 1 and back to 0 as 0 ≤ t ≤ π. Thus, when t goes from 0 to π, the points on the line segment will be traced out twice.

Step 4: When 0 ≤ t ≤ 2π

When the t-range is changed to 0 ≤ t ≤ 2π, we can see that the sine function goes from 0 to 1 and back to 0 twice during this interval. This means that the line segment will be traced out four times as the sine function completes two full cycles within the given range.

Key Concepts

Precalculus

Precalculus serves as the foundation for all advanced mathematics, especially calculus. It covers a variety of mathematical concepts that prepare students for the study of calculus and other higher-level math courses.

Precalculus topics often include functions, polynomials, rational functions, logarithms, and trigonometry. A thorough understanding of precalculus principles can help students make sense of more complex mathematical procedures later on, such as finding a parameterization of a line segment.

Parameterization itself is a precalculus concept where we represent a curve or line segment using equations that are dependent on a parameter, often represented by 't'. This becomes an essential skill in calculus for dealing with curves and surfaces in multiple dimensions.

Trigonometric Functions

Trigonometric functions are a vital component of both precalculus and calculus and are essential in modeling periodic phenomena. They relate the angles of a triangle to the lengths of its sides and are defined for all real numbers when they are treated as functions on a unit circle.

The primary trigonometric functions include sine (sin), cosine (cos), and tangent (tan), along with their reciprocals cosecant (csc), secant (sec), and cotangent (cot). These functions are used to describe wave patterns, circular motion, and many other natural phenomena and have applications in various fields including physics, engineering, and astronomy.

How Trigonometric Functions Relate to Parameterization

When parameterizing a line segment, trigonometric functions can be used if we want to express the line segment in such a way that reflects certain properties, such as its symmetry or periodicity.

Sine Function Properties

The sine function is one of the principal trigonometric functions and has some unique properties that make it useful in various mathematical applications.

• The domain of the sine function is all real numbers, and its range is between -1 and 1.
• It is an odd function, which means that \(\sin(-\theta) = -\sin(\theta)\).
• It is periodic, with a period of \(2\pi\), which means that its values repeat at regular intervals of \(2\pi\).

In parameterizing a line segment, these properties of the sine function can be utilized to control how the segment is traced as the parameter changes. By considering these properties, you can predict the behavior of a point moving along the segment as the parameter, in this case 't', varies.

Period of Sine Function

Understanding the period of the sine function is crucial when dealing with trigonometric equations and parameterizations involving trigonometry. The period of a trigonometric function is the length of the smallest interval over which the function's values repeat.

For the sine function, the standard period is \(2\pi\), but this can be modified by adjusting the function's input. For example, stretching or compressing the graph horizontally. In the context of parameterizing a line segment with sine functions, knowing that the sine completes a full cycle over an interval of \(2\pi\) allows us to predict and control the repetition of the line segment's tracing.

Application in Parameterizing a Line

If the parameter 't' ranges from 0 to \(2\pi\), you know a point will trace the line segment a certain amount of times based on how many multiples of \(2\pi\) fit into that range, which directly relates to how often the sine function repeats its values.